Somewhat like the 'burning to death, screaming' wiggle that we suffer after getting targeted by the bot-owned, laser-guided molotovs.
I think such death animations would add a lot to the atmosphere of the game. It's one thing to see a guy with his legs blown off and another one to see him on his back, grasping his stubs while bleeding out, screaming. It's one thing to see a bunch of bodies in a room, and another to see a bunch of dying people, moaning, calling for help, calling for their mothers, insulting the people that killed them.
The bleeding out shouldn't take for ever of course. But a random timer would be great, leaving some fallen soldiers or insurgents still grasping to life long after the battle, abandoned by his killers as they move on to the next objective.
This last thing would be abused by players, tho at least in coop such abuse isn't a problem: Players could also remain in the death throes, keeping their first-person perspective and VOIP location, still communicating with their team. It would be a source of great jokes and silly RP, on top of an authentic and awesome experience. And when it comes to multiplayer and team VOIP/discord communication in that state - shooting the downed player again should speed up the bleeding. AND this would also give a moment for the players to exchange pleasantries, be it a moan, congratulation on the kill, accusation of cheating or an insult.
I'll add: This is one of those things that always seemed obvious to me, but no game has ever done. This would make Sandstorm really stand out.