Currently Dominion does have a very significant first player advantage, and it would be nice if there were some reasonable way to fix it. Equal turns doesn't quite do that, but I think it would make things closer. It would however mean the first player could rarely trigger the endgame with a guaranteed win, but normally they'd be alright.
I think there
is a way to remove the "extra turn" advantage without giving the last player an end-game advantage, when players count points during the game:
Play equal turns, but after the ending condition was triggered, a player may not gain more points in his "extra" turn than the "triggering player" did in his last turn
* - or rather, any surplus points gained don't count for the score.
So in a 2P game, if the first player ends the game, he can essentially choose whether the final score, or the score after the penultimate round, counts.
This way, each player can end the game with a guaranteed win (the first player by ending the game when leading
before their last turn, the last player in the usual way). The first player can also "risk" to end the game without a secure win, e.g. by buying the last province or doing a megaturn. The PPR remains in force for both players (though with different numbers for first/second player).
I haven't tried it yet, but this rule should substantially reduce the first-player advantage, though not remove it completely.
*and may not remove VPs from him (due to cursers, trashing attacks etc.)