I wrote a program; I like to know what I'm getting into before trying to prove things. I can already suggest that it appears all Hurwitz quaternions are expressible as
$$ q = u (r^2 + s^2),   $$
where $q,u,r,s$ are Hurwitz quaternions and $u$ is a unit (there are 24 of those). For any fixed norm, so far at least one out of six such quaternions are the sum of two squares, and multiplication on the left by a unit gives everything. The worst behavior is norm 21, only 136 out of 768 Hurwitz quaternions of norm 21 are the sum of two squares.

About the sizes of things, in finding the sums of two squares (before multiplying by any unit): so far it has sufficed to take $r,s$ with norms less than double the norm of $q,$ so the norms of their squares are less that 4 times the square of the norm  of $q.$ The extremes for this seem to occur when the norm of $q$ is already the square of a prime $p \equiv 3 \pmod 4.$ So, two examples are $q = r^2 + s^2$ with
$$     q =  \frac{  5}{2 }   - \frac{ 3 }{2 } i   - \frac{ 1 }{2 } j  - \frac{ 1 }{2 } k, \; \;              r^2=   \frac{  19}{2 }   + \frac{ 21 }{2 } i   + \frac{ 7 }{2 } j  + \frac{ 7 }{2 } k, \; \;       s^2 = -7 - 12 i - 4 j - 4 k  $$
with norms $9,225,225,$ and $225 / 81 \approx 2.777$

Next $q = r^2 + s^2$ with
$$     q =  \frac{  13}{2 }   - \frac{ 5 }{2 } i   - \frac{ 1 }{2 } j  - \frac{ 1 }{2 } k, \;  $$
$$             r^2=   -\frac{  61}{2 }   - \frac{ 165 }{2 } i   - \frac{ 33 }{2 } j  - \frac{ 33 }{2 } k, \;        s^2 = 37 +80 i +16 j +16 k  $$
with norms $49,8281,8281,$ and $8281 / 2401 \approx 3.449$

EDIT, Sunday: I ran norm 121 overnight out of curiousity, with bound $6 n^2,$ which may or may not have really been large enough to correctly count the two squares. An extreme was $q = r^2 + s^2$ with
$$     q =  \frac{  7}{2 }   + \frac{ 11 }{2 } i   + \frac{ 1 }{2 } j  - \frac{ 17 }{2 } k, \;  $$
$$             r^2=   \frac{  407}{2 }   + \frac{ 155 }{2 } i   + \frac{ 341 }{2 } j  + \frac{ 31 }{2 } k, \;  \; \;      s^2 = -200 -72 i -168 j -24 k  $$
with norms $121,76729,73984,$ and $76729 / 14641 \approx 5.241$
   

If need be I can find out what the squares are squares of. Part of a big speed improvement was dropping that printout. 


Alright, sssssatistics. As I said, before multiplying on the left by the 24 units, the sums of two squares are not usually all items of that norm. For example, in norm $1,$ the six Hurwitz quaternions $\pm i, \pm j, \pm k$ are not the sum of two squares. Not my fault. 


      norm     two squares   not         total 
       1          18           6          24
       2           6          18          24
       3          68          28          96
       4          24           0          24
       5          84          60         144
       6          24          72          96
       7         144          48         192
       8          18           6          24
       9         162         150         312
      10          42         102         144
      11         180         108         288
      12          88           8          96
      13         228         108         336
      14          48         144         192
      15         432         144         576
      16          24           0          24
      17         180         252         432
      18          84         228         312
      19         392          88         480
      20         120          24         144
      21         136         632         768   
      22         120         168         288
      23         432         144         576
      24          96           0          96
      25         390         354         744
      26          90         246         336
      27         724         236         960
      28         184           8         192
      29         276         444         720
      30         192         384         576
      31         600         168         768
      32          24           0          24
      33         564         588        1152
      34         114         318         432
      35         864         288        1152
      36         240          72         312
      37         564         348         912
      38         168         312         480
      39         848         496        1344
      40         138           6         144
      41         588         420        1008
      42         192         576         768  
      43         792         264        1056  
      44         264          24         288    
      45         900         972        1872      
      46         216         360         576     
      47         936         216        1152    
      48          88           8          96    
      49         642         726        1368    
  

....................................................