Document:

Exhibit 10.4

 

July
26, 2021

 

Alpha
Healthcare Acquisition Corp. III

1177
Avenue of the Americas, 5th Floor

New
York, New York 10036

 

	Re:	Initial
    Public Offering

 

Ladies
and Gentlemen:

 

This
letter (this “Letter Agreement”) is being delivered to you in accordance with the Underwriting
Agreement (the “Underwriting Agreement”) entered into by and among Alpha Healthcare Acquisition Corp.
III, a Delaware corporation (the “Company”), and BofA Securities, Inc. and PJT Partners LP, as representatives
(the “Representatives”) of the several underwriters (each, an “Underwriter” and collectively,
the “Underwriters”), relating to an underwritten initial public offering (the “Public Offering”),
of 15,000,000 of the Company’s units (including up to 2,250,000 units that may be purchased to cover over-allotments, if any) (the
“Units”), each comprised of one share of the Company’s Class A common stock, par value $0.0001 per share
(the “Common Stock”), and one-fourth of one redeemable warrant. Each whole warrant (each,
a “Warrant”) entitles the holder thereof to purchase one share of Common Stock at a price of $11.50 per share,
subject to adjustment. The Units will be sold in the Public Offering pursuant to a registration statement on Form S-1 (File No. 333-253876)
and prospectus (the “Prospectus”) filed by the Company with the U.S. Securities and Exchange Commission (the
“Commission”) and the Company has applied to have the Units listed on The Nasdaq Capital Market. Certain capitalized
terms used herein are defined in paragraph 11 hereof.

 

In
order to induce the Company and the Underwriters to enter into the Underwriting Agreement and to proceed with the Public Offering and
for other good and valuable consideration, the receipt and sufficiency of which are hereby acknowledged, each of AHAC Sponsor III LLC
(the “Sponsor”) and the undersigned individuals, each of whom is a member of the Company’s board of directors
and/or management team or an advisor of the Company (each, an “Insider” and collectively, the “Insiders”),
hereby agrees with the Company as follows:

 

1.
The Sponsor and each Insider agrees that if the Company seeks stockholder approval of a proposed Business Combination, then in connection
with such proposed Business Combination, it, he or she shall (i) vote any shares of Capital Stock owned by it, him or her in favor of
any proposed Business Combination and (ii) not redeem any shares of Common Stock owned by it, him or her in connection with such stockholder
approval. If the Company engages in a tender offer in connection with any proposed Business Combination, the Sponsor and each Insider
agrees that it, he or she will not seek to sell or tender any shares of Capital Stock owned by it, him or her to the Company in connection
with such tender offer.

 

2.
The Sponsor and each Insider hereby agrees that in the event that the Company fails to consummate a Business Combination 24 months from
the closing of the Public Offering, or such later period approved by the Company’s stockholders in accordance with the Company’s
amended and restated certificate of incorporation (as it may be amended from time to time, the “Charter”),
the Sponsor and each Insider shall take all reasonable steps to cause the Company to (i) cease all operations except for the purpose
of winding up, (ii) as promptly as reasonably possible but not more than 10 business days thereafter, subject to lawfully available funds
therefor, redeem 100% of the shares of Common Stock sold as part of the Units in the Public Offering (the “Offering Shares”),
at a per-share price, payable in cash, equal to the aggregate amount then on deposit in the Trust Account (as defined below), including
interest earned on the funds held in the Trust Account and not previously released to the Company to pay its taxes (less up to $100,000
of interest to pay dissolution expenses), divided by the number of then outstanding Offering Shares, which redemption will completely
extinguish the rights of all holders of Offering Shares as stockholders (including the right to receive further liquidating distributions,
if any), subject to applicable law, and (iii) as promptly as reasonably possible following such redemption, subject to the approval of
the Company’s remaining stockholders and the Company’s board of directors, dissolve and liquidate, subject in the case of
clauses (ii) and (iii) to the Company’s obligations under Delaware law to provide for claims of creditors and the requirements
of other applicable law. The Sponsor and each Insider agrees not to propose any amendment to the Charter to modify (i) the substance
or timing of the ability of holders of Offering Shares to seek redemption in connection with the Company’s initial Business Combination
or amendments to the Charter prior thereto or (ii) (A) the Company’s obligation to redeem 100% of the Offering Shares if the Company
does not complete its initial Business Combination within such time set forth in the Charter or (B) with respect to any other provision
relating to stockholders' rights or pre-initial Business Combination activity, unless the Company provides the holders of the Offering
Shares with the opportunity to redeem their shares of Common Stock upon approval of any such amendment at a per-share price, payable
in cash, equal to the aggregate amount then on deposit in the Trust Account, including interest earned on the funds held in the Trust
Account and not previously released to the Company to pay its taxes, divided by the number of then outstanding Offering Shares.

 

     

     

    

 

        The
Sponsor and each Insider acknowledges that it, he or she has no right, title, interest or claim of any kind in or to any monies held
in the Trust Account or any other asset of the Company as a result of any liquidation of the Company with respect to the Founder Shares
or Private Placement Shares held by it, him or her. The Sponsor and each Insider hereby further waives, with respect to any shares of
Common Stock held by it, him or her, if any, whether acquired now or hereafter, any redemption rights it, he or she may have in connection
with the consummation of a Business Combination or amendments to the Charter prior thereto, including, without limitation, any such rights
available in the context of a stockholder vote to approve such Business Combination or a stockholder vote to approve an amendment to
the Charter to modify (i) (A) the substance or timing of the Company’s obligation to redeem 100% of the Offering Shares if the
Company has not consummated a Business Combination within the time period set forth in the Charter or (B) any other provisions relating
to stockholders’ rights or pre-initial Business Combination activity or (ii) in the context of a tender offer made by the Company
to purchase shares of Common Stock (although the Sponsor, the Insiders and their respective affiliates shall be entitled to redemption
and liquidation rights with respect to any Offering Shares it or they hold if the Company fails to consummate a Business Combination
within the time period set forth in the Charter).

 

3.
During the period commencing on the effective date of the Underwriting Agreement and ending 180 days after such date, the Sponsor and
each Insider shall not, without the prior written consent of the Representatives, (i) sell, offer to sell, contract or agree to sell,
hypothecate, pledge, grant any option to purchase or otherwise dispose of or agree to dispose of, directly or indirectly, or establish
or increase a put equivalent position or liquidate or decrease a call equivalent position within the meaning of Section 16 of the Securities
Exchange Act of 1934, as amended (the “Exchange Act”), and the rules and regulations of the Commission promulgated
thereunder, with respect to any Units, shares of Capital Stock, Warrants or any securities convertible into, or exercisable, or exchangeable
for, shares of Capital Stock owned by it, him or her, (ii) enter into any swap or other arrangement that transfers to another, in whole
or in part, any of the economic consequences of ownership of any Units, shares of Capital Stock, Warrants or any securities convertible
into, or exercisable, or exchangeable for, shares of Capital Stock owned by it, him or her, whether any such transaction is to be settled
by delivery of such securities, in cash or otherwise, or (iii) publicly announce any intention to effect any transaction specified in
clause (i) or (ii). Each of the Insiders and the Sponsor acknowledges and agrees that, prior to the effective date of any release or
waiver, of the restrictions set forth in this paragraph 3 or paragraph 7 below, the Company shall announce the impending release or waiver
by press release through a major news service at least two business days before the effective date of the release or waiver. Any release
or waiver granted shall only be effective two business days after the publication date of such press release. The provisions of this
paragraph will not apply if the release or waiver is effected solely to permit a transfer not for consideration and the transferee has
agreed in writing to be bound by the same terms described in this Letter Agreement to the extent and for the duration that such terms
remain in effect at the time of the transfer.

 

4.
In the event of the liquidation of the Trust Account upon the failure of the Company to consummate its initial Business Combination within
the time period set forth in the Charter, the Sponsor (the “Indemnitor”) agrees to indemnify and hold harmless
the Company against any and all loss, liability, claim, damage and expense whatsoever (including, but not limited to, any and all legal
or other expenses reasonably incurred in investigating, preparing or defending against any litigation, whether pending or threatened)
to which the Company may become subject as a result of any claim by (i) any third party for services rendered or products sold to the
Company or (ii) any prospective target business with which the Company has entered into a written letter of intent, confidentiality or
other similar agreement or Business Combination agreement (a “Target”); provided, however, that such
indemnification of the Company by the Indemnitor shall (x) apply only to the extent necessary to ensure that such claims by a third party
or a Target do not reduce the amount of funds in the Trust Account to below the lesser of (i) $10.00 per Offering Share and (ii) the
actual amount per Offering Share held in the Trust Account as of the date of the liquidation of the Trust Account, if less than $10.00
per Offering Share is then held in the Trust Account due to reductions in the value of the trust assets, less interest earned on the
Trust Account which may be withdrawn to pay taxes, (y) not apply to any claims by a third party or a Target which executed a waiver of
any and all rights to the monies held in the Trust Account (whether or not such waiver is enforceable) and (z) not apply to any claims
under the Company’s indemnity of the Underwriters against certain liabilities, including liabilities under the Securities Act of
1933, as amended. The Indemnitor shall have the right to defend against any such claim with counsel of its choice reasonably satisfactory
to the Company if, within 15 days following written receipt of notice of the claim to the Indemnitor, the Indemnitor notifies the Company
in writing that it shall undertake such defense.

  

    2

     

    

 

5.
To the extent that the Underwriters do not exercise their over-allotment option to purchase up to an additional 2,250,000 Units in full
within 45 days from the date of the Prospectus (and as further described in the Prospectus), the Sponsor agrees to forfeit, at no cost,
a number of Founder Shares in the aggregate equal to 562,500 multiplied by a fraction, (i) the numerator of which is 2,250,000 minus
the number of Units purchased by the Underwriters upon the exercise of their over-allotment option, and (ii) the denominator of which
is 2,250,000. The forfeiture will be adjusted to the extent that the over-allotment option is not exercised in full by the Underwriters
so that the Sponsor will be required to forfeit only that number of Founder Shares as is necessary so that the Initial Stockholders will
own an aggregate of 20.0% of the Company’s issued and outstanding shares of Capital Stock after the Public Offering (not including
the Private Placement Shares).

 

6.
The Sponsor and each Insider hereby agrees and acknowledges that: (i) the Underwriters and the Company would be irreparably injured in
the event of a breach by such Sponsor or an Insider of its, his or her obligations under paragraphs 1, 2, 3, 4, 5, 7(a), 7(b), and 9,
as applicable, of this Letter Agreement (ii) monetary damages may not be an adequate remedy for such breach and (iii) the non-breaching
party shall be entitled to injunctive relief, in addition to any other remedy that such party may have in law or in equity, in the event
of such breach.

 

7.
(a) The Sponsor and each Insider agrees that it, he or she shall not Transfer any Founder Shares (or shares of Common Stock issuable
upon conversion thereof) until the earlier to occur of (A) one year after the completion of the Company’s initial Business Combination
and (B) subsequent to the Company’s initial Business Combination, (x) if the reported last sale price of the Common Stock equals
or exceeds $12.00 per share (as adjusted for stock splits, stock dividends, reorganizations, recapitalizations and the like) for any
20 trading days within any 30-trading day period commencing at least 150 days after the Company’s initial Business Combination,
or (y) the date on which the Company completes a liquidation, merger, capital stock exchange, reorganization or other similar transaction
that results in all of the Company’s stockholders having the right to exchange their shares of Common Stock for cash, securities
or other property (the “Founder Shares Lock-up Period”).

 

(b)
The Sponsor and each Insider agrees that it, he or she shall not Transfer any Private Placement Units, the Private Placement Shares,
the Private Placement Warrants or shares of Common Stock issued or issuable upon the exercise of the Private Placement Warrants, until
30 days after the completion of the initial Business Combination (the “Private Placement Units Lock-up Period”,
together with the Founder Shares Lock-up Period, the “Lock-up Periods”).

 

(c)
Notwithstanding the provisions set forth in paragraphs 7(a) and 7(b), Transfers of the Founder Shares, Private Placement Units, Private
Placement Shares, Private Placement Warrants and shares of Common Stock issued or issuable upon the exercise or conversion of the Private
Placement Warrants or the Founder Shares that are held by the Sponsor, any Insider or any of their permitted transferees (that have complied
with this paragraph 7(c)), are permitted (i) to the Company’s officers or directors, any affiliate or family member of any of the
Company’s officers or directors or any members of the Sponsor or any affiliates of the Sponsor; (b) in the case of an individual,
by gift to a member of such individual’s immediate family or to a trust, the beneficiary of which is a member of such individual’s
immediate family, an affiliate of such individual or to a charitable organization; (c) in the case of an individual, by virtue of laws
of descent and distribution upon death of such individual; (d) in the case of an individual, pursuant to a qualified domestic relations
order; (e) by private sales or transfers made in connection with the consummation of an initial Business Combination at prices no greater
than the price at which the securities were originally purchased; (f) in the event of the Company’s liquidation prior to the completion
of an initial Business Combination; (g) by virtue of the laws of the State of Delaware or the Sponsor’s limited liability company
agreement upon dissolution of the Sponsor; or (h) in the event of the Company’s liquidation, merger, capital stock exchange,
reorganization or other similar transaction which results in all of the Company’s stockholders having the right to exchange their
shares of Common Stock for cash, securities or other property subsequent to the Company’s completion of an initial Business Combination;
provided, however, that in the case of clauses (a) through (e) or (g), these permitted transferees must enter into a written
agreement with the Company agreeing to be bound by the transfer restrictions herein and the other restrictions contained in this Agreement
(including provisions relating to voting, the Trust Account and liquidating distributions).

 

    3

     

    

 

8.
The Sponsor and each Insider represents and warrants that it, he or she has never been suspended or expelled from membership in any securities
or commodities exchange or association or had a securities or commodities license or registration denied, suspended or revoked. Each
Insider’s biographical information furnished to the Company (including any such information included in the Prospectus) is true
and accurate in all respects and does not omit any material information with respect to the Insider’s background. Each Insider’s
questionnaire furnished to the Company is true and accurate in all respects. Each Insider represents and warrants that: it, he or she
is not subject to or a respondent in any legal action for, any injunction, cease-and-desist order or order or stipulation to desist or
refrain from any act or practice relating to the offering of securities in any jurisdiction; it, he or she has never been convicted of,
or pleaded guilty to, any crime (i) involving fraud, (ii) relating to any financial transaction or handling of funds of another person,
or (iii) pertaining to any dealings in any securities and it, he or she is not currently a defendant in any such criminal proceeding.

 

9.
Except as disclosed in the Prospectus, neither the Sponsor nor any officer or director of the Company nor any affiliate of the Sponsor
nor any affiliate of any officer or director of the Company, shall receive from the Company any finder’s fee, reimbursement, consulting
fee, monies in respect of any repayment of a loan or other compensation prior to, or in connection with any services rendered in order
to effectuate, the consummation of the Company’s initial Business Combination (regardless of the type of transaction that it is),
, other than the following, none of which will be made from the proceeds held in the Trust Account prior to the completion of the initial
Business Combination: repayment of up to an aggregate of $300,000 in loans made to the Company by an affiliate of the Sponsor to cover
offering-related and organizational expenses; payment to an affiliate of the Sponsor of $10,000 per month, for up to 24 months, for office
space, utilities and secretarial and administrative support; reimbursement for any reasonable out-of-pocket expenses related to identifying,
investigating and completing an initial Business Combination; and repayment of non-interest bearing loans which may be made by the Sponsor
or an affiliate of the Sponsor or certain of the Company’s officers and directors to finance transaction costs in connection with
an intended initial Business Combination, the terms of which (other than as described above) have not been determined nor have any written
agreements been executed with respect thereto. Up to $1,500,000 of such loans may be convertible into units, at a price of $10.00 per
unit at the option of the lender, upon consummation of the initial Business Combination. The units would be identical to the Private
Placement Units.

 

10.
The Sponsor and each Insider has full right and power, without violating any agreement to which it is bound (including, without limitation,
any non-competition or non-solicitation agreement with any employer or former employer), to enter into this Letter Agreement and, as
applicable, to serve as an officer and/or director on the board of directors of the Company and hereby consents to being named in the
Prospectus as an officer and/or director of the Company.

 

11.
As used herein, (i) “Business Combination” shall mean a merger, capital stock exchange,
asset acquisition, stock purchase, reorganization or similar business combination, involving the Company and one or more businesses;
(ii) “Capital Stock” shall mean, collectively, the Common Stock and the Founder Shares;
(iii) “Founder Shares” shall mean (a) the 4,312,500 shares of the Company’s Class
B common stock, par value $0.0001 per share, initially issued to the Sponsor (up to 562,500 Shares of which are subject to complete or
partial forfeiture by the Sponsor if the over-allotment option is not exercised by the Underwriters) for an aggregate purchase price
of $25,000, or $0.0057971 per share, prior to the consummation of the Public Offering; (iv) “Initial Stockholders”
shall mean the Sponsor and any Insider that holds Founder Shares; (v) “Private Placement Shares”
shall mean the 455,000 shares of Common Stock comprising the Private Placement Units (or up to 500,000 shares of Common Stock if the
over-allotment option is exercised in full); (vi) “Private Placement Units”
shall mean the 455,000 units, or up to 500,000 units if the over-allotment option is exercised in full, each comprised of one share of
Common Stock and one-fourth of one warrant to purchase one share of Common Stock, that the Sponsor has agreed to purchase for an aggregate
purchase price of $4,550,000 (or up to $5,000,000 if the over-allotment option is exercised in full), or purchase price of $10.00 per
Private Placement Unit, in a private placement that shall occur simultaneously with the consummation of the Public Offering; (vii) “Private Placement Warrants”
shall mean the Warrants to purchase up to 113,750 shares of Common Stock (or up to 125,000 shares of Common Stock if the over-allotment
option is exercised in full) comprising the Private Placement Units; (viii) “Public Stockholders”
shall mean the holders of securities issued in the Public Offering; (ix) “Trust Account”
shall mean the trust fund into which a portion of the net proceeds of the Public Offering shall be deposited; and (x) “Transfer”
shall mean the (a) sale of, offer to sell, contract or agreement to sell, hypothecate, pledge, grant of any option to purchase or otherwise
dispose of or agreement to dispose of, directly or indirectly, or establishment or increase of a put equivalent position or liquidation
with respect to or decrease of a call equivalent position within the meaning of Section 16 of the Exchange Act, and the rules and regulations
of the Commission promulgated thereunder with respect to, any security, (b) entry into any swap or other arrangement that transfers to
another, in whole or in part, any of the economic consequences of ownership of any security, whether any such transaction is to be settled
by delivery of such securities, in cash or otherwise, or (c) public announcement of any intention to effect any transaction specified
in clause (a) or (b).

 

    4

     

    

 

12.
The Company will maintain an insurance policy or policies providing directors’ and officers’ liability insurance, and each
Director shall be covered by such policy or policies, in accordance with its or their terms, to the maximum extent of the coverage available
for any of the Company’s directors or officers.

 

13.
This Letter Agreement constitutes the entire agreement and understanding of the parties hereto in respect of the subject matter hereof
and supersedes all prior understandings, agreements, or representations by or among the parties hereto, written or oral, to the extent
they relate in any way to the subject matter hereof or the transactions contemplated hereby. This Letter Agreement may not be changed,
amended, modified or waived (other than to correct a typographical error) as to any particular provision, except by a written instrument
executed by all parties hereto.

 

14.
No party hereto may assign either this Letter Agreement or any of its rights, interests, or obligations hereunder without the prior written
consent of the other parties. Any purported assignment in violation of this paragraph shall be void and ineffectual and shall not operate
to transfer or assign any interest or title to the purported assignee. This Letter Agreement shall be binding on the Sponsor and each
Insider and their respective successors, heirs and assigns and permitted transferees.

 

15.
Nothing in this Letter Agreement shall be construed to confer upon, or give to, any person or corporation other than the parties hereto
any right, remedy or claim under or by reason of this Letter Agreement or of any covenant, condition, stipulation, promise or agreement
hereof. All covenants, conditions, stipulations, promises and agreements contained in this Letter Agreement shall be for the sole and
exclusive benefit of the parties hereto and their successors, heirs, personal representatives and assigns and permitted transferees.

 

16.
This Letter Agreement may be executed in any number of original or facsimile counterparts and each of such counterparts shall for all
purposes be deemed to be an original, and all such counterparts shall together constitute but one and the same instrument.

 

17.
This Letter Agreement shall be deemed severable, and the invalidity or unenforceability of any term or provision hereof shall not affect
the validity or enforceability of this Letter Agreement or of any other term or provision hereof. Furthermore, in lieu of any such invalid
or unenforceable term or provision, the parties hereto intend that there shall be added as a part of this Letter Agreement a provision
as similar in terms to such invalid or unenforceable provision as may be possible and be valid and enforceable.

 

18.
This Letter Agreement shall be governed by and construed and enforced in accordance with the laws of the State of New York, without giving
effect to conflicts of law principles that would result in the application of the substantive laws of another jurisdiction. The parties
hereto (i) all agree that any action, proceeding, claim or dispute arising out of, or relating in any way to, this Letter Agreement shall
be brought and enforced in the courts of New York City, in the State of New York, and irrevocably submit to such jurisdiction and venue,
which jurisdiction and venue shall be exclusive and (ii) waive any objection to such exclusive jurisdiction and venue or that such courts
represent an inconvenient forum.

 

19.
Any notice, consent or request to be given in connection with any of the terms or provisions of this Letter Agreement shall be in writing
and shall be sent by express mail or similar private courier service, by certified mail (return receipt requested), by hand delivery
or facsimile transmission.

 

20.
This Letter Agreement shall terminate on the earlier of (i) the expiration of the Lock-up Periods or (ii) the liquidation of the Company;
provided, however, that this Letter Agreement shall earlier terminate in the event that the Public Offering is not consummated and closed
by December 31, 2021; provided further that paragraph 4 of this Letter Agreement shall survive such liquidation.

 

21.
The Company, the Sponsor and each Insider hereby acknowledges and agrees that the Representatives on behalf of the Underwriters are each
a third party beneficiary of this Letter Agreement.

 

[Signature
Page Follows]

 

    5

     

    

  

	 	Sincerely,
	 	 
	 	AHAC
    SPONSOR III LLC
	 	 	 
	 	By:	 /s/
    Rajiv Shukla
	 	 	Name:
    Rajiv Shukla
	 	 	Title:   Managing
    Member

 

	 	By:	 /s/
    Rajiv Shukla
	 	 	Name:
    Rajiv Shukla

 

	 	By:	 /s/
    Patrick A. Sturgeon
	 	 	Name:
    Patrick A. Sturgeon

 

	 	By:	 /s/
    Darlene T. DeRemer
	 	 	Name:
    Darlene T. DeRemer

 

	 	By:	 /s/
    Eugene Podsiadlo 
	 	 	Name:
    Eugene Podsiadlo

 

	 	By:	 /s/
    William Woodward
	 	 	Name:
    William Woodward

   

	Acknowledged
    and Agreed:	 
	 	 
	ALPHA HEALTHCARE ACQUISITION CORP. III
	 	 	 
	By:	/s/
Rajiv Shukla	 
	 	Name: 	Rajiv Shukla	 
	 	Title:  	Chief Executive Officer	 

  

 

[Signature
Page to Letter Agreement]Exhibit 10.5

 

UNIT SUBSCRIPTION AGREEMENT

 

This UNIT SUBSCRIPTION AGREEMENT (this “Agreement”)
is made as of July 26, 2021, by and between Alpha Healthcare Acquisition Corp. III, a Delaware corporation (the “Company”),
having its principal place of business at 1177 Avenue of the Americas, 5th Floor, New York, New York 10036, and AHAC Sponsor
III LLC, a Delaware limited liability company (the “Subscriber”), having its principal place of business at 1177 Avenue
of the Americas, 5th Floor, New York, New York 10036.

 

WHEREAS, the Company desires to sell to the Subscriber
on a private placement basis (the “Placement”) an aggregate of 455,000 units (the “Initial Units”)
of the Company, and up to an additional 45,000 units (the “Additional Units” and, together with the Initial Units,
the “Units”) in the event that the underwriters’ 45-day over-allotment option (“Over-Allotment Option”)
is exercised in full or part, each Unit comprised of one share of Class A common stock of the Company, par value $0.0001 per share (“Common
Stock”) and one-fourth of one warrant, each whole warrant exercisable to purchase one share of Common Stock (“Warrant”),
for a purchase price of $10.00 per Unit. The shares of Common Stock underlying the Warrants are hereinafter referred to as the “Warrant
Shares”. The shares of Common Stock underlying the Units (excluding the Warrant Shares) are hereinafter referred to as
the “Placement Shares.” The Warrants underlying the Units are hereinafter referred to as the “Placement Warrants.” The
Units, Placement Shares, Placement Warrants and Warrant Shares, collectively, are hereinafter referred to as the “Securities.” Each
whole Placement Warrant is exercisable to purchase one share of Common Stock at an exercise price of $11.50 during the period commencing
30 days following the consummation of the Company’s initial business combination (the “Business Combination”),
as such term is defined in the registration statement in connection with the Company’s initial public offering of units (the “IPO”),
as amended at the time it becomes effective (the “Registration Statement”), and expiring on the fifth anniversary of
the consummation of the Business Combination; and

 

WHEREAS, the Subscriber wishes to purchase the
Initial Units and up to 45,000 Additional Units, and the Company wishes to accept such subscription from the Subscriber.

 

NOW, THEREFORE, in consideration of the premises
and the mutual covenants hereinafter set forth and other good and valuable consideration, the receipt and sufficiency of which are hereby
acknowledged, the Company and the Subscriber hereby agree as follows:

 

1.
Agreement to Subscribe

 

1.1. Purchase and Issuance of the Initial Units.
Upon the terms and subject to the conditions of this Agreement, the Subscriber hereby agrees to purchase from the Company, and the Company
hereby agrees to sell to the Subscriber, on the Initial Closing Date (as defined below) the Initial Units in consideration of the payment
of the Purchase Price (as defined below). On the Initial Closing Date, the Company shall, at its option, deliver to the Subscriber the
certificates representing the Securities purchased or effect such delivery in book-entry form.

 

1.2. Purchase Price. The Subscriber shall pay $4,550,000
(the “Purchase Price”) by wire transfer of immediately available funds or by such other method as may be reasonably
acceptable to the Company, to the trust account (the “Trust Account”) at a financial institution to be chosen
by the Company, maintained by Continental Stock Transfer & Trust Company, acting as trustee (“Continental”), one
(1) business day prior to the date of effectiveness of the Registration Statement.

 

1.3. Initial Closing. The closing of the purchase
and sale of 455,000 Initial Units shall take place simultaneously with the closing of the IPO (the “Initial Closing Date”).
The closing of such Units shall take place at the offices of Goodwin Procter LLP, The New York Times Building, 620 Eighth Avenue, New
York, New York, 10018, or such other place as may be agreed upon by the parties hereto.

 

1.4. Purchase and Issuance of Additional Units.
Upon the terms and subject to the conditions of this Agreement, the Subscriber hereby agrees to purchase from the Company, and the Company
hereby agrees to sell to the Subscriber, on the Over-allotment Closing Date (as defined below) up to an aggregate of 45,000 Additional
Units in consideration of the payment of $10.00 per Additional Unit for a purchase price of up to $450,000 and in the same proportion
as the amount of the Over-Allotment Option is exercised. On the Over-Allotment Closing Date (as defined below), the Company shall, at
its option, deliver to the Subscriber the certificates representing the Securities purchased or effect such delivery in book-entry form.

 

     

     

    

 

1.5. Purchase Price. As payment in full for the
Additional Units being purchased under this Agreement, the Subscriber shall pay $10.00 per Additional Unit being purchased by wire transfer
of immediately available funds or by such other method as may be reasonably acceptable to the Company, to the Trust Account on the date
of the consummation of the closing of the over-allotment option, and concurrently with the consummation thereof, or on such earlier time
and date as may be mutually agreed by the Company and the Subscriber (each such date, an “Over-Allotment Closing Date”;
together with the Initial Closing Date, the “Closing Dates” and each, a “Closing Date”).

 

1.6. Over-Allotment Closing. The Over-Allotment
Closing Date shall take place at the offices of Goodwin Procter LLP, The New York Times Building, 620 Eighth Avenue, New York, New York,
10018, or such other place as may be agreed upon by the parties hereto.

 

1.7 Termination. This Agreement and each of the
obligations of the undersigned shall be null and void and without effect if the Initial Closing Date does not occur prior to December
31, 2021.

 

2. Representations and Warranties of the
Subscriber

 

The Subscriber represents and warrants to the Company
that:

 

2.1. No Government Recommendation or Approval. The
Subscriber understands that no federal or state agency has passed upon or made any recommendation or endorsement of the Company or the
Placement of the Securities.

 

2.2. Accredited Investor. The Subscriber represents
that it is an “accredited investor” as such term is defined in Rule 501(a) of Regulation D under the Securities Act of 1933,
as amended (the “Securities Act”), and acknowledges that the sale contemplated hereby is being made in reliance, among
other things, on a private placement exemption to “accredited investors” under the Securities Act and similar exemptions under
state law.

 

2.3.  Intent. The Subscriber is purchasing
the Securities solely for investment purposes, for the Subscriber’s own account (and/or for the account or benefit of its members
or affiliates, as permitted, pursuant to the terms of an agreement (the “Insider Letter”) to be entered into with respect
to the Securities between, among others, the Subscriber and the Company, as described in the Registration Statement), and not with
a view to the distribution thereof and the Subscriber has no present arrangement to sell the Securities to or through any person or entity
except as may be permitted under the Insider Letter. The Subscriber shall not engage in hedging transactions with regard to the Securities
unless in compliance with the Securities Act.

 

2.4.  Restrictions on Transfer. The Subscriber
acknowledges and understands the Units are being offered in a transaction not involving a public offering in the United States within
the meaning of the Securities Act. The Securities have not been registered under the Securities Act and, if in the future the Subscriber
decides to offer, resell, pledge or otherwise transfer the Securities, such Securities may be offered, resold, pledged or otherwise transferred
only (A) pursuant to an effective registration statement filed under the Securities Act, (B) pursuant to an exemption from registration
under Rule 144 promulgated under the Securities Act, if available, or (C) pursuant to any other available exemption from the registration
requirements of the Securities Act, and in each case in accordance with any applicable securities laws of any state or any other jurisdiction.
Notwithstanding the foregoing, the Subscriber acknowledges and understands the Securities are subject to transfer restrictions as described
in Section 8 hereof. The Subscriber agrees that if any transfer of its Securities or any interest therein is proposed to be made, as a
condition precedent to any such transfer, the Subscriber may be required to deliver to the Company an opinion of counsel satisfactory
to the Company with respect to such transfer. Absent registration or another available exemption from registration, the Subscriber agrees
it will not resell the Securities (unless otherwise permitted pursuant to the Insider Letter, as described in the Registration Statement). The
Subscriber further acknowledges that because the Company is a shell company, Rule 144 may not be available to the Subscriber for the resale
of the Securities until the one year anniversary following consummation of the initial Business Combination of the Company, despite technical
compliance with the requirements of Rule 144 and the release or waiver of any contractual transfer restrictions.

 

    2

     

    

 

2.5. Sophisticated Investor.

 

(i) The Subscriber is sophisticated in financial
matters and is able to evaluate the risks and benefits of the investment in the Securities.

 

(ii) The Subscriber is aware that an investment
in the Securities is highly speculative and subject to substantial risks because, among other things, the Securities are subject to transfer
restrictions and have not been registered under the Securities Act and therefore cannot be sold unless subsequently registered under the
Securities Act or an exemption from such registration is available. The Subscriber is able to bear the economic risk of its investment
in the Securities for an indefinite period of time.

 

2.6. Independent Investigation. The Subscriber,
in making the decision to purchase the Units, has relied upon an independent investigation of the Company and has not relied upon any
information or representations made by any third parties or upon any oral or written representations or assurances from the Company, its
officers, directors or employees or any other representatives or agents of the Company, other than as set forth in this Agreement. The
Subscriber is familiar with the business, operations and financial condition of the Company and has had an opportunity to ask questions
of, and receive answers from the Company’s officers and directors concerning the Company and the terms and conditions of the offering
of the Units and has had full access to such other information concerning the Company as the Subscriber has requested. The Subscriber
confirms that all documents that it has requested have been made available and that the Subscriber has been supplied with all of the additional
information concerning this investment which the Subscriber has requested.

 

2.7  Organization and Authority. The
Subscriber is duly organized, validly existing and in good standing under the laws of the State of Delaware and it possesses all requisite
power and authority necessary to carry out the transactions contemplated by this Agreement.

 

2.8. Authority. This Agreement has been validly
authorized, executed and delivered by the Subscriber and is a valid and binding agreement enforceable in accordance with its terms, subject
to the general principles of equity and to bankruptcy or other laws affecting the enforcement of creditors’ rights generally.

 

2.9. No Conflicts. The execution, delivery
and performance of this Agreement and the consummation by the Subscriber of the transactions contemplated hereby do not violate, conflict
with or constitute a default under (i) the Subscriber’s charter documents, (ii) any agreement or instrument to which the Subscriber
is a party or (iii) any law, statute, rule or regulation to which the Subscriber is subject, or any agreement, order, judgment or decree
to which the Subscriber is subject.

 

2.10. No Legal Advice from Company. The
Subscriber acknowledges it has had the opportunity to review this Agreement and the transactions contemplated by this Agreement and the
other agreements entered into between the parties hereto with the Subscriber’s own legal counsel and investment and tax advisors. Except
for any statements or representations of the Company made in this Agreement and the other agreements entered into between the parties
hereto, the Subscriber is relying solely on such counsel and advisors and not on any statements or representations of the Company or any
of its representatives or agents for legal, tax or investment advice with respect to this investment, the transactions contemplated by
this Agreement or the securities laws of any jurisdiction.

 

2.11. Reliance on Representations and Warranties. The
Subscriber understands the Units are being offered and sold to the Subscriber in reliance on exemptions from the registration requirements
under the Securities Act, and analogous provisions in the laws and regulations of various states, and that the Company is relying upon
the truth and accuracy of the representations, warranties, agreements, acknowledgments and understandings of the Subscriber set forth
in this Agreement in order to determine the applicability of such provisions.

 

2.12. No General Solicitation. The Subscriber
is not subscribing for the Units as a result of or subsequent to any general solicitation or general advertising, including but not limited
to any advertisement, article, notice or other communication published in any newspaper, magazine, or similar media or broadcast over
television or radio, or presented at any seminar or meeting or in a registration statement with respect to the IPO filed with the Securities
and Exchange Commission (“SEC”).

 

    3

     

    

 

2.13. Legend. The Subscriber acknowledges
and agrees the certificates (if any) evidencing each of the Securities shall bear a restrictive legend (the “Legend”),
in form and substance substantially as set forth in Section 4 hereof.

 

3. Representations, Warranties and Covenants
of the Company

 

The Company represents and warrants to, and agrees
with, the Subscriber that:

 

3.1.  Valid Issuance of Capital Stock. The
total number of shares of all classes of capital stock which the Company has authority to issue is 100,000,000 shares of Class A Common
Stock, 10,000,000 shares of Class B Common Stock, $0.0001 par value per share (the “Class B Common Stock”), and 1,000,000
shares of preferred stock, $0.0001 par value per share (“Preferred Stock”). As of the date hereof, the Company has
issued and outstanding 4,312,500 shares of Class B Common Stock (of which up to 562,500 shares are subject to forfeiture as described
in the Registration Statement), no shares of Class A Common Stock and no shares of Preferred Stock. All of the issued shares of capital
stock of the Company have been duly authorized, validly issued, and are fully paid and non-assessable.

 

3.2.  Title to Securities. Upon issuance
in accordance with, and payment pursuant to, the terms hereof and that certain warrant agreement to be entered into between the Company
and Continental, as warrant agent (the “Warrant Agreement”), as the case may be, each of the Units, Placement Shares,
Placement Warrants and Warrant Shares will be duly and validly issued, fully paid and non-assessable. On the date of issuance of the Units,
the Warrant Shares shall have been reserved for issuance. Upon issuance in accordance with, and payment pursuant to, the terms hereof
and the Warrant Agreement, as the case may be, the Subscriber will have or receive good title to the Units, Placement Shares and Placement
Warrants, free and clear of all liens, claims and encumbrances of any kind, other than (i) transfer restrictions hereunder and pursuant
to the Insider Letter and (ii) transfer restrictions under federal and state securities laws.

 

3.3. Organization and Qualification. The Company
is a corporation duly incorporated, validly existing and in good standing under the laws of the State of Delaware and has the requisite
corporate power to own its properties and assets and to carry on its business as now being conducted.

 

3.4. Authorization; Enforcement. (i) The
Company has the requisite corporate power and authority to enter into and perform its obligations under this Agreement and to issue the
Securities in accordance with the terms hereof, (ii) the execution, delivery and performance of this Agreement by the Company and
the consummation by it of the transactions contemplated hereby have been duly authorized by all necessary corporate action, and no further
consent or authorization of the Company or its Board of Directors or stockholders is required, and (iii) this Agreement constitutes
valid and binding obligations of the Company enforceable against the Company in accordance with its terms, except as such enforceability
may be limited by applicable bankruptcy, insolvency, fraudulent conveyance, moratorium, reorganization, or similar laws relating to, or
affecting generally the enforcement of, creditors’ rights and remedies or by equitable principles of general application and except
as enforcement of rights to indemnity and contribution may be limited by federal and state securities laws or principles of public policy.

 

3.5. No Conflicts. The execution, delivery
and performance of this Agreement and the consummation by the Company of the transactions contemplated hereby do not (i) result in
a violation of the Company’s certificate of incorporation or by-laws, (ii) conflict with, or constitute a default under any
agreement or instrument to which the Company is a party or (iii) any law statute, rule or regulation to which the Company is subject or
any agreement, order, judgment or decree to which the Company is subject. Other than any SEC or state securities filings which may be
required to be made by the Company subsequent to a Closing Date, and any registration statement which may be filed pursuant thereto, the
Company is not required under federal, state or local law, rule or regulation to obtain any consent, authorization or order of, or make
any filing or registration with, any court or governmental agency or self-regulatory entity in order for it to perform any of its obligations
under this Agreement or issue the Units, Placement Shares, Placement Warrants or Warrant Shares in accordance with the terms hereof.

  

    4

     

    

 

4. Legends

 

4.1. Legend. The Company will issue the Units,
Placement Shares and Placement Warrants, and when issued, the Warrant Shares, purchased by the Subscriber in the name of the Subscriber.
The certificates (if any) evidencing Securities will bear the following Legend and appropriate “stop transfer” instructions:

 

“THE SECURITIES REPRESENTED HEREBY
HAVE NOT BEEN REGISTERED UNDER THE SECURITIES ACT OF 1933, AS AMENDED (THE “SECURITIES ACT”), OR ANY STATE SECURITIES LAWS
AND NEITHER THE SECURITIES NOR ANY INTEREST THEREIN MAY BE OFFERED, SOLD, TRANSFERRED, PLEDGED OR OTHERWISE DISPOSED OF EXCEPT PURSUANT
TO AN EFFECTIVE REGISTRATION STATEMENT UNDER THE SECURITIES ACT OR SUCH LAWS OR AN EXEMPTION FROM REGISTRATION UNDER THE SECURITIES ACT
AND SUCH LAWS WHICH, IN THE OPINION OF COUNSEL FOR THIS CORPORATION, IS AVAILABLE.”

 

“THE SECURITIES REPRESENTED BY
THIS CERTIFICATE ARE SUBJECT TO LOCKUP PURSUANT TO AN INSIDER LETTER BETWEEN, AMONG OTHERS, ALPHA HEALTHCARE ACQUISITION CORP. III AND
AHAC SPONSOR III LLC AND MAY ONLY BE OFFERED, SOLD, TRANSFERRED, PLEDGED OR OTHERWISE DISPOSED DURING THE TERM OF THE LOCKUP PURSUANT
TO THE TERMS SET FORTH IN THE INSIDER LETTER.”

 

4.2.  Subscriber’s Compliance. Nothing
in this Section 4 shall affect in any way the Subscriber’s obligations and agreements to comply with all applicable securities
laws upon resale of the Securities.

 

4.3.  Company’s Refusal to Register
Transfer of the Securities. The Company shall refuse to register any transfer of the Securities, if in the sole judgment of the Company
such purported transfer would not be made (i) pursuant to an effective registration statement filed under the Securities Act, or
pursuant to an available exemption from the registration requirements of the Securities Act and (ii) in compliance herewith and with the
Insider Letter.

 

4.4.  Registration Rights. The Subscriber
will be entitled to certain registration rights which will be governed by a registration rights agreement (“Registration Rights
Agreement”) to be entered into between, among others, the Subscriber and the Company, on or prior to the effective date of the
Registration Statement. 

 

5. Waiver of Liquidation Distributions.

 

In connection with the Securities purchased pursuant
to this Agreement, the Subscriber hereby waives any and all right, title, interest or claim of any kind in or to any distributions of
the amounts in the Trust Account with respect to the Securities, whether (i) in connection with the exercise of redemption rights if the
Company consummates the Business Combination, (ii) in connection with any tender offer conducted by the Company prior to a Business Combination,
(iii) upon the Company’s redemption of shares of Common Stock sold in the Company’s IPO upon the Company’s failure to
timely complete the Business Combination or (iv) in connection with a stockholder vote to approve an amendment to the Company’s
amended and restated certificate of incorporation (A) to modify the substance or timing of the Company’s obligation to redeem 100%
of the Company’s public shares if the Company does not timely complete the Business Combination or (B) with respect to any other
provision relating to stockholders’ rights or pre-Business Combination activity. In the event the Subscriber purchases shares
of Common Stock in the IPO or in the aftermarket, any additional shares so purchased shall be eligible to receive the redemption value
of such shares of Common Stock upon the same terms offered to all other purchasers of Common Stock in the IPO in the event the Company
fails to consummate the Business Combination.

 

6. Terms of Placement Warrants. Each
Placement Warrant shall have the terms set forth in the Warrant Agreement.

 

    5

     

    

 

7. Rescission Right Waiver and Indemnification.

 

7.1. The Subscriber understands and acknowledges
an exemption from the registration requirements of the Securities Act requires there be no general solicitation of purchasers of the Units.
In this regard, if the IPO were deemed to be a general solicitation with respect to the Units, the offer and sale of such Units may not
be exempt from registration and, if not, the Subscriber may have a right to rescind its purchase of the Units. In order to facilitate
the completion of the Placement and in order to protect the Company, its stockholders and the amounts in the Trust Account from claims
that may adversely affect the Company or the interests of its stockholders, the Subscriber hereby agrees to waive, to the maximum extent
permitted by applicable law, any claims, right to sue or rights in law or arbitration, as the case may be, to seek rescission of its purchase
of the Units. The Subscriber acknowledges and agrees this waiver is being made in order to induce the Company to sell the Units to the
Subscriber. The Subscriber agrees the foregoing waiver of rescission rights shall apply to any and all known or unknown actions, causes
of action, suits, claims or proceedings (collectively, “Claims”) and related losses, costs, penalties, fees, liabilities
and damages, whether compensatory, consequential or exemplary, and expenses in connection therewith, including reasonable attorneys’
and expert witness fees and disbursements and all other expenses reasonably incurred in investigating, preparing or defending against
any Claims, whether pending or threatened, in connection with any present or future actual or asserted right to rescind the purchase of
the Units hereunder or relating to the purchase of the Units and the transactions contemplated hereby.

 

7.2. The Subscriber agrees not to seek recourse
against the Trust Account for any reason whatsoever in connection with its purchase of the Units or any Claim that may arise now or in
the future.

 

7.3. The Subscriber acknowledges and agrees that
the stockholders of the Company are and shall be third-party beneficiaries of this Section 7.

 

7.4. The Subscriber agrees that to the extent any
waiver of rights under this Section 7 is ineffective as a matter of law, the Subscriber has offered such waiver for the benefit of the
Company as an equitable right that shall survive any statutory disqualification or bar that applies to a legal right. The Subscriber acknowledges
the receipt and sufficiency of consideration received from the Company hereunder in this regard.

 

8. Terms of the Units and Placement Warrants

 

8.1. The Units and their component parts are substantially
identical to the units to be offered in the IPO except that: (i) the Units and component parts are subject to the transfer restrictions
described in the Insider Letter, (ii) the Placement Warrants will be non-redeemable if called for redemption pursuant to Section 6.1 of
the Warrant Agreement so long as they are held by the Subscriber (or any of its permitted transferees) and as otherwise provided in Section
5 herein, and may be exercisable on a “cashless” basis if held by the Subscriber or its permitted transferees, as further
described in the Warrant Agreement and (iii) the Units and component parts are being purchased pursuant to an exemption from the registration
requirements of the Securities Act and will become freely tradable only after the expiration of the lockup described above in clause (i)
and they are registered pursuant to the Registration Rights Agreement to be signed on or before the date of the Prospectus or an exemption
from registration is available.

 

8.2 The Subscriber agrees to vote the Placement
Shares in accordance with the terms of the Insider Letter and as otherwise described in the Registration Statement.

 

9. Governing Law; Jurisdiction; Waiver
of Jury Trial

 

This Agreement shall be governed by and construed
in accordance with the laws of the State of New York for agreements made and to be wholly performed within such state. The
parties hereto hereby waive any right to a jury trial in connection with any litigation pursuant to this Agreement and the transactions
contemplated hereby.

 

    6

     

    

 

10. Assignment; Entire Agreement; Amendment

 

10.1.  Assignment. Neither this Agreement
nor any rights hereunder may be assigned by any party to any other person other than by the Subscriber to a person agreeing to be bound
by the terms hereof, including the waiver contained in Section 7 hereof.

 

10.2.  Entire Agreement. This Agreement sets
forth the entire agreement and understanding between the parties as to the subject matter thereof and merges and supersedes all prior
discussions, agreements and understandings of any and every nature among them.

 

10.3. Amendment. Except as expressly provided
in this Agreement, neither this Agreement nor any term hereof may be amended, waived, discharged or terminated other than by a written
instrument signed by all of the parties hereto.

 

10.4.  Binding upon Successors. This Agreement
shall be binding upon and inure to the benefit of the parties hereto and to their respective heirs, legal representatives, successors
and permitted assigns. 

 

11. Notices

 

11.1  Notices. Unless otherwise provided herein,
any notice or other communication to a party hereunder shall be sufficiently given if in writing and personally delivered or sent by facsimile
or other electronic transmission with copy sent in another manner herein provided or sent by courier (which for all purposes of this Agreement
shall include Federal Express or other recognized overnight courier) or mailed to said party by certified mail, return receipt requested,
at its address provided for herein or such other address as either may designate for itself in such notice to the other. Communications
shall be deemed to have been received when delivered personally, on the scheduled arrival date when sent by next day or 2nd-day courier
service, or if sent by facsimile upon receipt of confirmation of transmittal or, if sent by mail, then three days after deposit in the
mail. If given by electronic transmission, such notice shall be deemed to be delivered (a) if by electronic mail, when directed to
an electronic mail address at which the stockholder has consented to receive notice; (b) if by a posting on an electronic network
together with separate notice to the stockholder of such specific posting, upon the later of (1) such posting and (2) the giving
of such separate notice; and (c) if by any other form of electronic transmission, when directed to the stockholder.

 

12. Counterparts

 

This Agreement may be executed in one or more counterparts,
all of which when taken together shall be considered one and the same agreement and shall become effective when counterparts have been
signed by each party and delivered to the other party, it being understood that both parties need not sign the same counterpart. In
the event that any signature is delivered by facsimile transmission or by e-mail delivery of a “pdf” format data file, such
signature shall create a valid and binding obligation of the party executing (or on whose behalf such signature is executed) with the
same force and effect as if such facsimile or “.pdf” signature page were an original thereof.

 

13. Survival; Severability

 

13.1.  Survival. The representations, warranties,
covenants and agreements of the parties hereto shall survive the Closing Dates.

 

13.2. Severability. In the event that any provision
of this Agreement becomes or is declared by a court of competent jurisdiction to be illegal, unenforceable or void, this Agreement shall
continue in full force and effect without said provision; provided that no such severability shall be effective if it materially changes
the economic benefit of this Agreement to any party.

 

14. Headings.

 

The titles and subtitles used in this Agreement
are used for convenience only and are not to be considered in construing or interpreting this Agreement.

 

    7

     

    

 

IN WITNESS WHEREOF, the parties hereto have executed
this Agreement to be effective as of the date first set forth above.

 

	 	COMPANY:
	 	 
	 	ALPHA HEALTHCARE ACQUISITION CORP. III
	 	 
	 	By: 	/s/ Patrick A. Sturgeon
	 	Name:  	 Patrick A. Sturgeon
	 	Title: 	Chief Financial Officer
	 	 
	 	SUBSCRIBER:
	 	 
	 	AHAC SPONSOR III LLC
	 	 
	 	By: 	/s/ Rajiv Shukla
	 	Name:   	Rajiv Shukla
	 	Title: 	Managing Member

 

[Unit Subscription Agreement with Sponsor]

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