Dogod: I once played this game in a correspondence club, as you described, except that there was no restriction on how far a friendly piece could be repelled. One useful tactic was positioning a piece behind a passed pawn, thereby pushing to the eighth rank for an immediate promotion! It was especially effective by moving a Rook or Queen, so that the newly gained Queen was defended.

Years ago, I belonged to a correspondence club which tried out many different ideas for CVs. One of the better ones we discovered and played was "MAGNETIC CHESS". The rules are:
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The object is to CAPTURE the enemy King. (The concept of 'check' does not exist here in the legal sense.) A move (whether capturing or not) by any unit other than a King generates an imaginary magnetic impulse ("magnetism") along horizontal and vertical axes of the landing square that repels friendly units (except the King) to the farthest open square on that axis and draws in opposing units (except the King) to an adjacent square. For example, White could open 1.c4, pulling Black's pawn to c5, 1.c4(c7c5). Black could then move the Queen to a5, creating two magnetic effects, 1. ... Qa5(a2a4,c5h5). (Note that the black pawn at a7 is unaffected because it is already next to Ra8.) In this way, each move involves the rearrangement of anywhere from one to five units, but magnetism extends only as far as open spaces allow.

Kings are not affected by magnetism, nor can magnetism affect any unit in a line beyond either King. King moves do not create magnetism, but during castling a Rook move does.

No en passant capturing, as it really does not make sense in this game concept. Friendly pawns may be repelled to the first or eighth rank. Pawns on the first or second rank may always make a two-step move, even if previously moved. A friendly Pawn on an unobstructed file propelled to the eighth rank is immediately promoted; this is most effectively accomplished by moving a Queen or Rook behind the Pawn (so that the new piece is defended).
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If you favor this idea and would like to play it, write to Fencer and/or post here. It would be a while before he could get it on the site, but the more requests he receives, the sooner it will happen.