No. Is not possible to play the same My Hospital game on your mobile device and on your computer.

If you download the game on your computer and connect to Game Center/Google Play, you will only be able to see your friends that are also playing My Hospital on their computers and also connected.

When you connect to Game Center/Google Play and you are playing on your phone, you are able to play the same game on another mobile device and you will be able to see your friends that are also playing on mobile.

The game in your computer and the game on your mobile device will always be 2 separate games and is not possible to connect them.