The folks at Eastmans’ Hunting Journals decided to test that idea in their recent video. The underwater archer in the video even demonstrates some reasonable accuracy by popping balloons with arrows at short distances. Obviously the accuracy would be hurt a lot at longer distances, but it is theoretically possible to hit a longer distance target underwater - provided you can still see it.
Bows are not designed to be fully submerged or fired underwater, but that doesn't mean you cannot still do it anyway. While doing this could potentially damage a wooden bow, using a compound bow means that you are highly unlikely to damage the bow since it doesn't rust and there is no wood that could become soggy and rot.
The biggest difference is that the arrows "swim" a lot slower than they normally fly due to the water resistance. But hey, at least there is no wind to mess up your shot, right? Just water currents if any.
