This got some discussion in here, so I thought I'd break it out and give it it's own thread. Basically, we were talking about another variant that uses the four-player version as the starting point but then adds some random events to drive everyone crazy. Probably not worth having ratings for it, as there would be a lot of randomness, like playing MarioKart on frantic settings.
1. Play until your king is captured or otherwise killed, ignoring check and checkmate.
2. After a random number of moves (3-6), a nuclear missile explodes on a random square. Any piece on that square is killed, and the square is irradiated for the next three turns; no pieces can move there or pass through. If it hits your king, tough luck, you're dead. Then the timer is reset to repeat this after another 3-6 moves.
3. At the start of the game, 4-6 random squares have hidden land mines placed on them. If a piece stops on a land mine, it is killed, and a new land mine is placed on a new random square.
4. If you get a pawn to the eight rank, you are not required to promote it. You can keep pushing it if you want. If you get a pawn all the way across the board, you can promote it to a second king so that you can survive the death of your first king.
5. A computer-controlled knight is placed in the middle of the board at the start of the game. After the fourth person's turn, it moves randomly. This knight cannot be captured or killed, and can survive nuclear missiles. It can jump off the board, and if it does, it is replaced back on an empty square near the middle of the board. (Note: see #9 as an alternative to this rule)
6. There are hidden clocks placed on several random squares. If one of your pieces stops on one of these, you get ten seconds added to your clock, and a new hidden clock is placed on another random square. There is no guarantee that a clock and a land mine cannot be on the same square.
7. When your king is killed, you get to fire a single nuclear missile at any square you choose; you cannot target an opponent's king.
8. At random intervals, a random piece of a random colour is changed to a random piece of a random colour. So a red bishop might turn into a green pawn. If this takes your king, too bad, you lose. If this gives you an extra king, you've got insurance against losing one of them.
9. (from @BabYagun, an alternative to #5) We should have a huge boss king in the middle of the board taking 4 squares (2x2). Every 10 moves the boss king jumps to some random place and crushes all the pieces below. It can be killed, but it has 5 lives.
9.a. (from @Skeftomilos) To allow you to influence the movement of the boss king, he should be attracted to queens (thus you put your queen near someone else's army to encourage him to move that direction), or repulsed by queens (you put your queen near him to chase him away), or his moves should be determined by a vote of all four members.
10. Anything else we can come up with.