Bare-knuckle boxing matches.
Although seriously, this is something that I've been thinking a bit about lately. A mode where there's no tricks. No specialties. Nothing. Just two users duking it out with nothing but their fists. Okay, well, I guess a pot or two might be in order, but for those who want the excitement of just the fight itself, I think this could possibly suffice. Not to mention it could create an entirely different kind of atmosphere for a fight.
Then again, I branched another idea off of this one.
Sumo matches. Two combatants fight on a circular arena mat and they try to knock each other off. Whoever falls off first loses. Not sure if something like that would have to be best 2 out of 3, but hey. This one would also be very bare-to-the-bones fighting. If we go with something like this, we might actually have to disable taking damage, but still allow knocking back.
Or there could be a combination of the two, Sumo Boxing. One player tries to either kill the other, or at least knock them off the mat. Just a couple of ideas I've had swimming around in my brain. I've made a Sumo Boxing ring in Hyrule, and I've been itching to make it a real thing.
EDIT: Well now. Guess I forgot to check and see if this was for ideas in general or for 2V2. I go sit in shame corner now.