AnarchyChess

Oh yeah, I like that.

I'd say it should do away with checkmate and use a Drawback Chess–style system where you win by capturing the king. This would cost 3 actions. I think merely moving into check could cost 2.

My original idea was that capturing with king would cost less than a normal capture because I wanted to buff the king's ability to protect himself, and in particular I was worried about strategies that could force the opponent to spend 2 actions capturing your piece with their king. But making moving into check cost 2 actions and capturing the king cost 3 would completely negate the need for that.

Moving into check with a capture would, obviously, cost 3 actions (1 for move, 1 extra for capturing, 1 extra for the check). And discovered checks would also cost 2 actions, with discovered check via a capture costing 3.