First i want to give a BIG thanks to softaculous for AMPPS!
I am not 100% sure if this is enough for you, but this is my way to hide the AMPPS Window:
My Setup: OS X 10.9.3 (Mavericks) & AMPPS 2.3
1. Open AMPPS
2. Click the red close Button
-> AMPPS will ask you: "Are you sure you want to quit AMPPS?"
3. press ESC
4. Window is gone but AMPPS is still running. And the window is coming back when you focus the App through CMD+TAB or by clicking the dock icon.