Elder Sign is nifty. It's... I don't know if "complicated" is the right word--detailed? Characters actually have hit points. Two different kinds of 'em.
If you've got role players, though, it should be fine. It does technically have player elimination that you said you don't like, but if it comes up the game is almost certainly almost over (and you're losing).
First the investigators try to make sure the Ancient Once stays sealed away by locating the Elder Signs. It's all but impossible to lose here (if an investigator is slain, the player just draws a new one), and if you collect enough you win. If you fail you get a chance to fight and kill it, but it's pretty hopeless and you've probably lost.
Each Ancient One has a different power that changes how the game is played even before they awaken. And some of 'em awaken quicker.
Legendary, the Mavel Deckbuilding Game has both cooperative and competitive elements. It's pricey, though, and I don't think I recommend it quite as highly. Unless the group's into Marvel, of course.
The players' goal as a group is to thwart the villainous mastermind's (Magneto, Red Skull, Dr. Doom, or Loki) plot and defeat him. They win if they beat him up. They lose if the plot's "Evil Wins" condition comes up. They thwart the villain, but fail to capture him, if the villain deck runs out without either happening.
Players also collect Victory Points for everything to accomplish, so trying to one-up each other is a secondary goal. There are cards that can either help or hinder the other players--help them when the team is losing, hinder them when you're winning and it's all about the VP.
I've picked up Flash Point recently, but haven't played it as much yet.
I do also recommend Pandemic.