Document:

EX-4.1

 Exhibit 4.1 

UNITS 
 NUMBER 

U- 
 SEE REVERSE FOR CERTAIN DEFINITIONS 

CUSIP 
 HUDSON EXECUTIVE
INVESTMENT CORP. III 
 UNITS CONSISTING OF ONE SHARE OF CLASS A COMMON STOCK AND ONE-FIFTH OF
ONE 
 REDEEMABLE WARRANT, EACH WHOLE WARRANT ENTITLING THE HOLDER TO PURCHASE 

ONE SHARE OF CLASS A COMMON STOCK 
 THIS
CERTIFIES THAT is the owner of Units. 
 Each Unit (“Unit”) consists of one share of Class A common stock, par
value $0.0001 per share (“Common Stock”), of Hudson Executive Investment Corp. III, a Delaware corporation (the “Company”), and one-fifth of one redeemable
warrant (the “Warrant”). Each whole Warrant entitles the holder to purchase one share of Common Stock for $11.50 per share (subject to adjustment). Only whole Warrants are exercisable. Each whole Warrant will become
exercisable on the later of (i) thirty (30) days after the Company’s completion of a merger, capital stock exchange, asset acquisition, stock purchase, reorganization or other similar business combination with one or more businesses (each
a “Business Combination”), and (ii) twelve (12) months from the closing of the Company’s initial public offering, and will expire unless exercised before 5:00 p.m., New York City Time, on the date that is five
(5) years after the date on which the Company completes its initial Business Combination, or earlier upon redemption or liquidation (the “Expiration Date”). The Common Stock and Warrants comprising the Units represented
by this certificate are not transferable separately prior to [ ], unless [•] and [•] elect to allow separate trading earlier, subject to the Company’s filing of a Current Report on Form 8-K with
the Securities and Exchange Commission containing an audited balance sheet reflecting the Company’s receipt of the gross proceeds of the Company’s initial public offering and issuing a press release announcing when separate trading will
begin. No fractional Warrants will be issued upon separation of the Units. The terms of the Warrants are governed by a Warrant Agreement, dated as of [ ], between the Company and Continental Stock Transfer & Trust Company, as Warrant Agent,
and are subject to the terms and provisions contained therein, all of which terms and provisions the holder of this certificate consents to by acceptance hereof. Copies of the Warrant Agreement are on file at the office of the Warrant Agent at 1
State Street, 30th Floor, New York, New York 10004, and are available to any Warrant holder on written request and without cost. 
 This
certificate is not valid unless countersigned by the Transfer Agent and registered by the Registrar of the Company. 
 This certificate
shall be governed by and construed in accordance with the internal laws of the State of New York. 
 Witness the facsimile signature of a
duly authorized signatory of the Company. 
  

					
	     
	  		  	     

			
	Authorized Signatory	  		  	Transfer Agent

 Hudson Executive Investment Corp. III 

The Company will furnish without charge to each unitholder who so requests, a statement of the powers, designations, preferences and relative,
participating, optional or other special rights of each class of equity or series thereof of the Company and the qualifications, limitations, or restrictions of such preferences and/or rights. 

The following abbreviations, when used in the inscription on the face of this certificate, shall be construed as though they were written out
in full according to applicable laws or regulations: 
  

					
	TEN COM — as tenants in common	  		  	UNIF GIFT MIN ACT—________Custodian________
			
	TEN ENT — as tenants by the entireties	  		  	                                      
      (Cust)                     (Minor)
			
	 JT TEN — as joint tenants with right of survivorship

and not as tenants in common
	  		  	                                    under Uniform
Gifts to Minors Act
		  		  	                                      
                                         
                         
			
		  		  	  
 (State)

 Additional abbreviations may also be used though not in the above list. 

For value received, hereby sell, assign and transfer unto 

(PLEASE INSERT SOCIAL SECURITY OR OTHER IDENTIFYING NUMBER OF ASSIGNEE) 

(PLEASE PRINT OR TYPEWRITE NAME AND ADDRESS, INCLUDING ZIP CODE, OF ASSIGNEE) 

Units represented by the within Certificate, and do hereby irrevocably constitute and appoint 

Attorney to transfer the said Units on the books of the within named Company with full power of substitution in the premises. 

 

			
	Dated	  	  
 Notice: The signature to this
assignment must correspond
 with the name as written upon the face of the certificate in every particular, without alteration or enlargement or any change
whatever.

  

			
	 Signature(s) Guaranteed:
	 	
		
	  

THE SIGNATURE(S) MUST BE GUARANTEED BY AN
	 	
	 ELIGIBLE GUARANTOR INSTITUTION (BANKS,
	 	
	 STOCKBROKERS, SAVINGS AND LOAN ASSOCIATIONS
	 	
	 AND CREDIT UNIONS WITH MEMBERSHIP IN AN
	 	
	 APPROVED SIGNATURE GUARANTEE MEDALLION
	 	
	 PROGRAM, PURSUANT TO S.E.C. RULE 17Ad-15 UNDER
	 	
	 THE SECURITIES EXCHANGE ACT OF 1934, AS
	 	
	 AMENDED (OR ANY SUCCESSOR RULE)).
	 	

 As more fully described in, and subject to the terms and conditions described in, the Company’s final
prospectus for its initial public offering dated [    ], the holder(s) of this certificate shall be entitled to receive a pro rata portion of certain funds held in the trust account established in connection with the
Company’s initial public offering only in the event that (i) the Company redeems the shares of Common Stock sold in the Company’s initial public offering and liquidates because it does not consummate an initial business combination by
the date set forth (the “Last Date”) in the Company’s Amended and Restated Certificate of Incorporation, as the same may be amended from time to time (the “Charter”), (ii) the Company redeems the shares of Common Stock sold
in its initial public offering properly submitted in connection with a stockholder vote to amend the Charter to modify the substance or timing of the Company’s obligation to redeem 100% of the Common Stock if it does not consummate an initial
business combination by the Last Date or with respect to any other material provisions relating to stockholders’ rights or pre-initial business combination activity, or (iii) if the holder(s) seek(s)
to redeem for cash his, her or its respective shares of Common Stock in connection with a tender offer (or proxy solicitation, solely in the event the Company seeks stockholder approval of the proposed initial business combination) setting forth the
details of a proposed initial business combination. In no other circumstances shall the holder(s) have any right or interest of any kind in or to the trust account.EX-4.2

 Exhibit 4.2 
  

			
	NUMBER	  	SHARES
	C-	  	

 SEE REVERSE FOR CERTAIN DEFINITIONS 

CUSIP 
 HUDSON EXECUTIVE
INVESTMENT CORP. III 
 CLASS A COMMON STOCK 

THIS CERTIFIES THAT is the owner of fully paid and non-assessable shares of Class A common stock,
par value $0.0001 per share (the “Common Stock”), of Hudson Executive Investment Corp. III, a Delaware corporation (the “Company”), transferable on the books of the Company in person or by duly
authorized attorney upon surrender of this certificate properly endorsed. 
 This certificate is not valid unless countersigned by the
Transfer Agent and registered by the Registrar of the Company. 
 Witness the facsimile signature of a duly authorized signatory of the
Company. 
  

					
		  		  	
	  
 Authorized Signatory
	  		  	  
 Transfer Agent

 HUDSON EXECUTIVE INVESTMENT CORP. III 

The Company will furnish without charge to each stockholder who so requests, a statement of the powers, designations, preferences and
relative, participating, optional or other special rights of each class of equity or series thereof of the Company and the qualifications, limitations, or restrictions of such preferences and/or rights. This certificate and the shares represented
thereby are issued and shall be held subject to all the provisions of the Company’s Amended and Restated Certificate of Incorporation and all amendments thereto and resolutions of the Board of Directors providing for the issue of securities
(copies of which may be obtained from the secretary of the Company), to all of which the holder of this certificate by acceptance hereof assents. 

The following abbreviations, when used in the inscription on the face of this certificate, shall be construed as though they were written out
in full according to applicable laws or regulations: 
  

															
	TEN COM	 	—	 	as tenants in common	  	UNIF GIFT MIN ACT—                            	  	Custodian	  	                                
		 		 		  		  	(Cust)        	  		  	(Minor)
							
	TEN ENT	 	—	 	as tenants by the entireties	  		  		  		  	
					
	JT TEN	 	—	 	as joint tenants with right of survivorship	  		  	under Uniform Gifts to Minors Act
		 		 	and not as tenants in common	  		  	  

		 		 		 		  		  		  	(State)	  	

 Additional abbreviations may also be used though not in the above list. 

For value received, hereby sells, assigns and transfers unto 

(PLEASE INSERT SOCIAL SECURITY OR OTHER IDENTIFYING NUMBER(S) OF ASSIGNEE(S)) 

(PLEASE PRINT OR TYPEWRITE NAME(S) AND ADDRESS(ES), INCLUDING ZIP CODE, OF ASSIGNEE(S)) 

shares of Common Stock represented by the within Certificate, and hereby irrevocably constitutes and appoints 

Attorney to transfer the said shares of Common Stock on the books of the within named Company with full power of substitution in the premises.

 Dated: 
  

	
	  

	Notice: The signature(s) to this assignment must correspond with the name as written upon the face of the certificate in every particular, without alteration or enlargement or any change whatever.

  

	
	 Signature(s) Guaranteed:

	THE SIGNATURE(S) MUST BE GUARANTEED BY AN ELIGIBLE GUARANTOR INSTITUTION (BANKS, STOCKBROKERS, SAVINGS AND LOAN ASSOCIATIONS AND CREDIT UNIONS WITH MEMBERSHIP IN AN APPROVED SIGNATURE GUARANTEE MEDALLION PROGRAM, PURSUANT TO S.E.C.
RULE 17Ad-15 UNDER THE SECURITIES EXCHANGE ACT OF 1934, AS AMENDED (OR ANY SUCCESSOR RULE).

 As more fully described in, and subject to the terms and conditions described in, the Company’s final
prospectus for its initial public offering dated [    ], the holder(s) of this certificate shall be entitled to receive a pro rata portion of certain funds held in the trust account established in connection with the
Company’s initial public offering only in the event that (i) the Company redeems the shares of Common Stock sold in the Company’s initial public offering and liquidates because it does not consummate an initial business combination by
the date set forth (the “Last Date”) in the Company’s Amended and Restated Certificate of Incorporation, as the same may be amended from time to time (the “Charter”), (ii) the Company redeems the shares of Common Stock sold
in its initial public offering properly submitted in connection with a stockholder vote to amend the Charter to modify the substance or timing of the Company’s obligation to redeem 100% of the Common Stock if it does not consummate an initial
business combination by the Last Date or with respect to any other material provisions relating to stockholders’ rights or pre-initial business combination activity, or (iii) if the holder(s) seek(s)
to redeem for cash his, her or its respective shares of Common Stock in connection with a tender offer (or proxy solicitation, solely in the event the Company seeks stockholder approval of the proposed initial business combination) setting forth the
details of a proposed initial business combination. In no other circumstances shall the holder(s) have any right or interest of any kind in or to the trust account.

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