What if I pass focus and my opponent passes focus in showdown

It closes the Showdown — when all players pass Focus in sequence without playing anything, the Showdown ends (Rule 348, see also 347.2.a).

What happens next depends on the type. If it was a Combat Showdown, you proceed to the Combat Damage Step and then Combat Cleanup (which heals all units in play). If it was a Non-Combat Showdown, no damage step and no healing: if only one player's units remain at the battlefield, that player Establishes Control and Conquers if they haven't already scored it this turn (Rule 348.2).

One nuance: the pass that matters is passing Focus while the chain is empty, and it takes all players passing in sequence. A pass after a chain resolves is just Focus auto-moving to the next player in turn order — that's not the same as a player passing, so the Showdown doesn't end there (see FAQ 1797).