h4 is definitely a theoretical playable move, but it will never transpose to nimzowitsch because the pawn is on c6 and not the knight, especially not when the pawn structure is blocked like this
Nimzowitch 4.h4?!

I would have thought it depended on your long term plan, for example whether or not you are planning a kingside invasion, and it is usually initiated by h4

In this particular position I think Black can play 4...f6. If White has to play 5.Nf3 then the pawn on h4 looks silly. If 5.f4 then 5...Nh6 hits the hole on g4.

With no pawn break (e.g. a pawn on g6) and no development I don't think 4.h4 in the caro plans anything other than gaining more space on that side. Space that may not even be useful until the late endgame. Plus I think it's only a reasonable move because it forces a pawn move from black in response (otherwise it just loses time in the opening).
But definitely I'm not sure. We've all seen h4 in the caro a million times, so it's expected, but a priori it strikes me as an odd move.
In the Caro-Kann it's for straight up attacking. The move 4.h4 prevents Black from continuing development with 4...e6 since 5.g4 will win the bishop. So Black plays either 4...h6 or 4...h5 to deal with this threat. 4...h5 5.Bg5 (preventing ...e6 again) 5...Qb6 and 4...h6 5.g4 Bd7 (5...Bh7 6.e6!) 6.h5 both lead to wild games. Tal popularized this move although he never played this 5.Bg5 line.
I have seen the move 4.h4!? in Caro-Kann. Do you think this can be formatted into Nimzowitch Defence (1.e4 Nc6) ?(One of the main lines is e4 Nc6 d4 d5 e5 Bf5). I am a Nimzowitch Player myself, so it would be useful to know if 4.h4!? can be played. (The point of h4 is that after 4...e6?? White can trap Black's bishop with 5.g4!!).
What do you think?