I still have the old version of the Anima cardgame.
And i was wondering what exactly is the new crisis system for the 'endgame'?

I don't know how the basic/final missions worked for the base game, but...

After any player has completed one of their basic missions, they can, at the start of any turn, decide to reveal the final mission. Once the mission is revealed, a turn counter starts. Some missions have a shorter timer than others (I think there's two 8-turn missions and one 10-turn in the base game, don't know about the expansion). Only players that have completed a basic mission can attempt the final mission. If the timer runs out before anyone completes the mission, everyone gets one last chance to win by completing the "crisis" version of the mission. Everyone that hasn't completed at least one basic mission immediately loses, then everyone else gets to try (in speed order, I think). If you fail, you lose the game. Yes, this does mean that you might have a game where no one wins.