I like this more than the handstand push-up. It requires a bit more upper chest musculature, plus the head can travel below the level of the hands, which makes for a deep press.
The Suicide Push-Up
Compared to shoulder presses, it's much easier to decelerate the lift's eccentric/lowering component. That ensures healthy shoulders.
The goal is to attain a pike position and "dive" downward, so you're looking right below your own feet in the bottom position. Press towards your butt. That means you won't press completely vertically but on a slight angle to incorporate the shoulders and upper chest together.
Controlled sets of 10-15 reps can be a game-changer for your shoulder health and development. Bonus: You'll get a great pump.