<PAGE>   1
 
                                                                    EXHIBIT 99.4
 
                          AMERICA WEST AIRLINES, INC.
 
                               OFFER TO EXCHANGE
                   PASS THROUGH CERTIFICATES, SERIES 1998-1,
                           WHICH HAVE BEEN REGISTERED
                 UNDER THE SECURITIES ACT OF 1933, AS AMENDED,
                          FOR ANY AND ALL OUTSTANDING
                    PASS THROUGH CERTIFICATES, SERIES 1998-1
 
To Our Clients:
 
     Enclosed for your consideration is a Prospectus of America West Airlines,
Inc., a Delaware corporation (the "Company" or "America West"), dated
            , 1999 (the "Prospectus"), and the enclosed Letter of Transmittal
(the "Letter of Transmittal") relating to the offer to exchange (the "Exchange
Offer") the registered Pass Through Certificates, Series 1998-1 (the "New
Certificates") for any and all outstanding Pass Through Certificates, Series
1998-1 (the "Old Certificates"), upon the terms and subject to the conditions
described in the Prospectus. The Exchange Offer is being made in order to
satisfy certain obligations of Continental contained in the Exchange and
Registration Rights Agreement dated as of October 6, 1998, between the Company
and the Initial Purchasers.
 
     The CUSIP numbers for the Old Certificates are as follows: Class A (023654
AK 2), Class B (023654 AL O), and Class C (023654 AM 8).
 
     This material is being forwarded to you as the beneficial owner of the Old
Certificates carried by us in your account but not registered in your name. A
tender of such Old Certificates may only be made by us as the holder of record
and pursuant to your instructions.
 
     Accordingly, we request instructions as to whether you wish us to tender on
your behalf the Old Certificates held by us for your account, pursuant to the
terms and conditions set forth in the enclosed Prospectus and Letter of
Transmittal.
 
     Your instructions should be forwarded to us as promptly as possible in
order to permit us to tender the Old Certificates on your behalf in accordance
with the provisions of the Exchange Offer. The Exchange Offer will expire at
5:00 p.m., New York City time, on             , 1999 (the "Expiration Date") (30
calendar days following the commencement of the Exchange Offer), unless extended
by the Company. Any Old Certificates tendered pursuant to the Exchange Offer may
be withdrawn at any time before 5:00 p.m., New York City time on the Expiration
Date.
 
     Your attention is directed to the following:
 
          1.  The Exchange Offer is for any and all Old Certificates.
 
          2.  The Exchange Offer is subject to certain conditions set forth in
     the Prospectus in the section captioned "The Exchange Offer -- Conditions".
 
          3.  Any transfer taxes incident to the transfer of Old Certificates
     from the holder to the Company will be paid by the Company, except as
     otherwise provided in the Instructions in the Letter of Transmittal.
 
          4.  The Exchange Offer expires at 5:00 p.m., New York City time, on
     the Expiration Date unless extended by the Company.
<PAGE>   2
 
     If you wish to have us tender your Old Certificates, please so instruct us
by executing and returning to us the instruction form set forth below. The
Letter of Transmittal is furnished to you for information only and may not be
used directly by you to tender Old Certificates.
 
                INSTRUCTIONS WITH RESPECT TO THE EXCHANGE OFFER
 
     The undersigned acknowledge(s) receipt of your letter enclosing the
Prospectus, dated             , 1998, of America West Airlines, Inc., a Delaware
corporation, and the related specimen Letter of Transmittal.
 
--------------------------------------------------------------------------------
 
     This will instruct you to tender the number of Old Certificates indicated
below held by you for the account of the undersigned, pursuant to the terms and
conditions set forth in the Prospectus and the related Letter of Transmittal.
(Check one).
 
<TABLE>
<S>   <C>          <C>
Box 1     [ ]      Please tender my Old Certificates held by you for my
                   account. If I do not wish to tender all of the Old
                   Certificates held by you, I have identified on a signed
                   schedule attached hereto the number of Old Certificates I do
                   not wish tendered.
 
Box 2     [ ]      Please do not tender any Old Certificates held by you for my
                   account.
</TABLE>
 
--------------------------------------------------------------------------------
 
<TABLE>
<S>                                          <C>
 
Date                , 199                    -----------------------------------------------------
                                                                 SIGNATURE(S)
                                             -----------------------------------------------------
                                             -----------------------------------------------------
                                                           PLEASE PRINT NAME(S) HERE
                                             -----------------------------------------------------
                                                          AREA CODE AND TELEPHONE NO.
</TABLE>
 
     UNLESS A SPECIFIC CONTRARY INSTRUCTION IS GIVEN IN THE SPACE PROVIDED, YOUR
                                  SIGNATURE(S)
  HEREON SHALL CONSTITUTE AN INSTRUCTION TO US TO TENDER ALL OLD CERTIFICATES.
 
                                        2