cool idea, I'm way too dumb to understand deeply, but it seems like it would be fun
Shadow clone chess (or ninjutsu chess)

@penguimms
if you have any questions, feel free to ask
I thought I made the explanation clear, but maybe some moments can be phrased better.

Cool idea, but it will be very hard to make and code. I think fog of war, and crazy house combined in the variants server is the best bet?

@jeshwia
Rook can either clone itself along the file or the rank (but not both at the same time). I guess, this restriction might be redundant, but probably helps reducing the complexity this rule introduce.
If a rook on b1 makes a clone on c1, second clone must go to d1. It's probably also ok to allow first clone to go to a1, and the second one to c1 (still didn't decide whether to allow this, probably should work just as fine).

Only two clones as OP says. It's possible to go for more, but then the whole board could get bloated and the game complexity will be insane.
Each piece can make two clones of itself, taking a whole turn. Clones must be placed on the adjacent squares (on the same vertical, horizontal or diagonal. The only exception is Knight, but more on this later, as the explanation would probably require to see this on a real board*) . If the adjacent squares are occupied, the move is considered illegal. Then player can choose which cloned piece he considers as "real" (let's call other clones "fakes"). If the "fake" piece gets attacked, it disappears by the next move . Similarly, if the "fake" King gets checked it also disappears. The "real" King might ignore checks, but if it gets captured, the side that captured King, wins. Each piece can make clones only once per the game. Pawns can't make clones.
* about Knights. Knight clones must be placed on the free squares that Knight can move to:
To avoid too much compications, clones of a Knight must be placed on the squares of the "same direction": if a Knight on g1 decides to place a clone on f3, the second clone must be placed on e5 . Second example: Black is threatening mate with the Bishop+Queen battery, but White replies with Nd2@c4@b6+ and wins Black's Queen.
This makes a Knight a long-range, and probably one of the most dangerous piece ever (after the Queen, probably).
UPD. Similarly, Bishops, can make only make clones along the same diagonals they move, the same rule applies to Rooks (they only make clones along the vertical or horizontal they move).
UPD2. Fake clones of the piece should be indistinguishable from the real piece for the opponent (at the same time, player should be able to distinguish fake clones of own pieces from the real piece).