I agree - I would like to add castling to this 4-player version of infinite chess.
Chess has the "quirky" rule of castling, and it would be fun to have it here too. When it happens within one army, here is the diagram how do it. The king moves 3 or 4 squares as you recommend. All intermediate squares need to be empty, and the rook finishes next to the king.
When castling with the teammate's rook, do you think all the intermediate squares need to be empty? That would be a fun maneuver because it would be one of the only moves where you move both your piece, and your teammate's piece in one turn. I like the idea, but just need to decided the rules for it.
I just thought of an idea for teams only.
Instead of each player having their own set of pieces, they share the team's set. For example, if someone on the white team uses the right pieces, they can also move the ones on the left and vice versa.
This would only work well if there is always consensus on the best move in a team.
Suppose I and vickalan are a team and I follow a certain strategy or tactic with "my" set of pieces and vickalan disagrees and makes a move with "my" set of pieces destroying my plans?
I think it would be better if each player has only his own set of pieces, but the two team members are allowed to coordinate with each other.