Multimove variant rules with queen that does two moves in one I recall playing a multimove variant special rules chess game where one player (say black) has all the usual pieces, while the other player has just a queen that can do two moves in one.

So the queen can strike one of black's pieces and return to its position as if nothing happened -- "action at a distance"!

It was such that if one player Alice was a bit more skilled than another Bob, then Alice could beat Bob whether playing white or black.

Anybody know of rules like these?
A variant that I have played for a while years ago is: white has just the king and the four central pawns (cdef), but is allowed two moves per turn. All usual rules, including en passant and promotion, apply. Check and checkmate are defined via "the player that takes the opposing king wins". The variant is surprisingly balanced, with black probably at a slight advantage (but with a lot more possibilities for blunders). 

Given that this variant is playable as it is, I suppose that anything involving a double-moving queen would be ridiculously overpowered for its side.

The white king in this variant can 'make a nearby piece disappear' as you mentioned. It takes a few games to understand the strategies for the two sides: against someone of comparable level who has never played this variant, a player with some experience can often ask them to pick a side, beat them, propose a rematch with sides switched, and then proceed to beat them again. :)