I like the idea of more than two teams c: but, what if, we did a type of spinoff of Ascension? I'll just try to build upon your idea if that's okay.
Maybe 6 3-man teams or 8 duos, or perhaps free for all? Each team (or player) gets points based on capturing some objective, which may just be collecting health packs or killing minions that pop up, while larger minions near the center award more points to whoever kills it, but with the drawback that there's more likelihood to run into someone near the center of the map?
Or maybe I shouldn't have watched The Hunger Games again xD
Or to stick with the four corner perspective, maybe the objective is to destroy/capture turrets to gain territory, and at the end of the game, the team with the largest territory wins? Let's say in each corner, there are six turrets that (somehow) divide the base into portions, and capturing one turret gives that portion to the team that captures it. But with this many players (I'd think 4 people per team would be ideal) how would champ select work? Would there be bans? Can people pick the same champ? And since this seems so crazy already, screen-lock might be required in order to be able to play properly? xD truly outrageous