Idea for someday in the far distant future: 4-player Crazyhouse

In four-player crazyhouse, the material gained from captures is probably enough to encourage aggressive play.  There's no disincentive to make a fair trade (in fact, you gain mobility because you've traded a bishop for a bishop that can be dropped anywhere), so I think the points system could be dispensed with and you could go to a straight-up last man standing system.

Because of the ease of promoting pawns, you might want to require them to get farther than the 8th rank before they're promoted, or you'll end up with 20 queens on the board in every game.  Since pawns can be dropped anywhere, maybe reaching the 11th rank is good enough, or maybe it should require all the way to the 14th; in that case only dropped pawns are likely to be promoted, but that's generally the case in crazyhouse anyway.

 Maybe four-player 960?

And king of the hill

4 player crazyhouse already exists

you can play it on judochess.com but there are no turns there

4 player crazyhouse sounds like a good idea to me!

If promoted queens were valued 9 points, the incentive to promote would be reduced significantly. Also the value of knights should be adjusted accordingly, since knight drops can be deadly! I would really like to play this variant!

I don't think a crazyhouse version should use points.  Points were only added because people were sitting back and waiting for everyone else to trade.  But since trading in theory strengthens your army (since your new piece can be dropped anywhere), I don't think we'll need a points system to encourage aggressive play.

Which means we should be able to go with last man standing instead of points.  That means we don't have to worry about piece values.

I like the introduction of points, because it makes the game feel like normal chess, where you play from start to end trying to build an advantage. In multi-player chess without points, the only thing you can build for the most part of the game is your social bonds with your occasional allies/enemies. Until two players remain, that is.

In crazyhouse, you can try to build an advantage even without points.  Every piece you win increases the size of your army.

The size of your army is not relevant in last-man-standing, until two players have been eliminated, or until you are so strong that you can beat all remaining opponents by yourself. The second case is almost impossible since they make more moves per turn than you, and even more in crazyhouse since the pieces you sack are reinforcing their armies.

Lets say that you managed to capture most of the army of an opponent, and then checkmated him. You just made yourself the target for a newly created alliance, that will gang up on you until you are again as weak as everybody else. Your early-game feats have left behind no memory. This is my understanding of the situation, and I am not thrilled about it!

Fair point - but once you are no stronger, you're no longer the target.  But actually, this might result in a permanent 3-way balance of power until someone loses on time.

Maybe we need a points system...

The problem with a points system is that it's effectively last man standing, as if you survive a long time you'll have traded a huge number of pieces to get there.

Maybe there's not a way to make this work...

Another thing: I can imagine that collusion in 4-players crazyhouse could be really nasty. If your king's defenses are not air-tight, two opponents should be able to achieve a coordinated checkmate against you out of nowhere. A single undefended square beside your king could by the cause of your elimination before your next turn. Or two attached empty squares, although well defended both of them, and -boooom!- you are dead!

That could be partially solved by playing until the king is captured instead of mate.  Then, while someone can drop a queen in front of your king and the next person can protect it, but you make the first person pay by taking their queen.

no points system in 4 way zh, does not make any sense, you could keep capturing an infinite amount of times

What about losing the points every time you drop a piece back in the board? This way you would increase your points by hoarding captured pieces in your reserve. You would drop pieces sparingly, and only when there was a promise for a greater material gain (or checkmate), or to avoid getting checkmated.

Capturing promoted queens would earn 9 points, and they would go to your reserve as pawns, but if you drop these pawns again you would lose 9 points.

Very fair

That might be a way to make it work.  I'm not entirely sure about your handling of promoted queens, but the general idea of gaining points for captures and losing points for drops seems sound.

That way, you can still gain points (up to a certain limit) if you capture more pieces than you lose - but you have an incentive to keep pieces in reserve.

It would be interesting to play-test this and see if you still tend to end up in a 3-way balance of power.

The simplest way to handle the promotions would be +1 point for capturing the promoted queen, and -1 for dropping back the pawn. But you know that I dislike the idea of 1 point valued queens! The drawback of the +9/-9 idea is that now there will be two types of pawns in your reserve. Pawns that cost -1 to drop, and pawns that cost -9 to drop, and designing the user interface accordingly could become challenging. Especially if promotions are not restricted to queens, but you are allowed to promote to other pieces as well.

As for the gameplay, I share your concerns about following the same patterns as the last-man-standing version. I can imagine that if you have obtained a solid lead in points, and as a result you are ganged up by the two remaining opponents, you would have the options either to defend your position by dropping pieces and losing your advantage, or resigning and hoping that your opponents will not outscore you.