This was already invented. In fact, it's actually called "Penguin Chess". Although I call it "Anywhere Chess".
Anywhere Chess: King may not teleport, pawns may not teleport to 1st or 8th rank.
Nearly Anywhere: You must make a normal move to check the opponent's king (directly or by discovery)
Parachute Anywhere: You must make a normal move to attack an opponent's piece (directly or by discovery)
Hardly Anywhere: In addition to the restrictions above, you may not teleport to block a check.
Me and some of my friends invented a new variant of chess (or at least I think it's new...) basically, every piece (but the king) can move to any empty space on the board. Weve been playing it for a few weeks and its really fun! The games mostly turn into a race to take f2/f7, but with so many ways to defend the games get really complex. Anyways, we've dubbed it 'Penguin chess', does anyone think it could be a valid variant to chess? :D