A Submersible aircraft carrier...?

Honestly, making it stealthy wouldn't be much of an issue, you can't hear our subs as is, and hearing is the only way to find things under water. Your idea is actually quite feasible, although would be astronomically expensive. To launch, you'd have to come to surface, which is just a matter of emptying some ballast tanks (all submersibles float when the ballast tanks are empty, obviously), and take off vertically.