As far as mechanics go, it would totally disrupt the game. It's a great idea, but its just not too feasible. If you had that ability, let's call it Rapid Swing for this discussion, you would totally dominate melee combat. At low levels, it could help, but mid to high levels, you would have a walking blender on your hands.
A 6th level fighter with that feat (and no bonuses, yeah right!), would have attacks at +6 and +1. With Rapid Swing, he would get attacks at +4, +4 and -1.
If that same fighter had two weapon fighting (and a light, off-hand weapon), he would get attacks at +2, +2, +2, and -3. And if for some reason that extra swing applied to his off-hand attacks as well, that would be +2, +2, -3 for primary and +0, +0 for his off-hand attacks. Having a 6th level anything making 5 attacks is silly.