As promised, I went and played a few games with the H4 opening. It was a while before I was able to get a good game.
Of course, this game started with me playing white and myself moving pawn to h4, and continued with g3 after my opponent moved d5.
The idea of the h4 opening is to get a powerful knight on the f4 square and a powerful bishop on the h3 or g4 square that cannot be kicked by a pawn. I developed my pieces with this idea in mind. My opponent had a very strong center that did not allow my pieces much movement or my knight to get to the f4 square, so I decided to open up the position with a pawn push in the center. My opponent castled in reply.
Castling in this position was a large blunder. I knew I could take advantage of this easily. With my king side pawns already extended dangerously close to his castle position, and my bishop in a fianchetto position with a diagonal also bearing down on his castle position, it would be easy to generate a vicious attack. Play continued, and after a few sacrifices to remove defenders and get my pieces in better places, the board looked like this.
I gave black the option to give me a strong double-pawn center or move his queen with my move knight to f4. He chose to move his queen to d6, putting some pressure on my d4 pawn while moving fleeing his queen. After, I opened my bishop on b2's scope by moving my pawn up one square to d5 with tempo after black moves his knight. After moving the knight, he threatened my queen with his bishop, which I replied to by moving my queen to d3 and threatening a mate. My opponent had no way to stop the mate without losing a large amount of material, so he resigned with the board looking like this.
Making a last, random, hopeless move just for fun with Nd4, my opponent resigned.