So currently, Azir has been in a weird place because his q can make his burst and early game too good which was not what Riot intended. I believe there is actually a somewhat simple solution. His ult, e, and passive all seem fine as they are to me.
Let's start with my biggest change, his q. In my opinion, his q should have its damage taken away completely while getting a bit of utility in compensation. Give his q charges (I'm thinking 3 would be fair possibly even going from 1-3 charges at ranks 1,3, and 5) with a fairly long cooldown, maybe 20 seconds. This should improve team fighting (late game) by letting him reposition 3 times to adjust based on what's happened rather than just one and done. However the long cooldown will mean he will still need to think about if using it will be worth having less charges if another team fight breaks out before the full minute required for 3 20 second charges is finished.
Next, Azir's w should be his main damage, not his soldiers, and currently they do quite well although sometimes I'm a bit unhappy with their range. Therefore, i don't want to touch them too much. However, to compensate for the hit to Azir's early game from removing his q damage, I believe his soldiers should have extended duration and range while under any allied towers (original and constructed). This will give Azir safer ability to farm under turret while still not buffing damage as I feel that would reduce too much counter play under his turrets. This is my first ever post, so I'm sure there's room for improvement. Just let me know.