I learned a checkers variant from the Mexican family across the street today. I don't remember what the name of the game was, though, because I couldn't remember a Spanish word if it bit me on the punta. If that meant something, it was an accident.
Anyhow, the two rules changes are quite simple. 1) The king moves like a bishop. He can capture again after making a capture, just like in normal checkers, but he can move (and capture) across great distance.
2) Forced captures. If you do not capture a piece when you have the opportunity, you lose the piece.
Rule two solves all the issues with checkers. The bishop rule keeps the game from getting slow in the endgame, but also simplifies it. I don't think it adds a whole lot, but I could be wrong.
The forced capture rule changes checkers from being brutally simplistic to being brutally complex. Suddenly, virtually every move forces another move, and you have to read five moves ahead instead of three.
Checkers is saved!
Okay, whatever. What you can take home from this demonstration is simple:
You don't need complex rules (or even a complex system) to make a game deep. If you make one move affect other moves intuitively, it makes the game deeper. If your game isn't deep enough, subtly change the rule set to futher entangle the moves.
For the capture, the only thing that changed was the rule for capture. It used to say:
You may capture any adjacent enemy piece with one of your pieces by "jumping" to the empty spot diagonally across it, if that is a direction you could usually move. You may string jumps together and capture multiple enemy pieces in this manner.
Now it says:
You must capture... You must string jumps... if you do not jump when you have the opportunity, that piece is removed from play.
Such small changes change the face of the game. They can change the face of your game, too.
Fun variant: Instead of kinging the piece you reach the end of the table with, you may king any of your pieces. Only non-kings can trigger a kingening.