What I'm imagining is that after each move, a chess engine (fritz or whatever) could run invisibly on the server to analyse the position in a game in progress, and then the players could see a number/rating to indicate who was "winning" at any given time.

Obviously it's not going to be super accurate, but still might be interesting? 

The problem with that is if one side makes a blunder, the opponent will know that and take his time to find the winning move.

Evaluating whether a position is winning or losing is one of the things the players are supposed to figure out. And yes, it would make spotting blunders and determining whether a sacrificial combination forces a win too easy.

It could be cool to have available for your games archieve. But maybe it would take to much computing power to analyse every single game played on chess.com using an engine, to be a viable feature...

Post-analysis, as fluffy_rabbit suggests, is more reasonable. :)

That's an interesting idea, but it would make it much easier to cheat.  Maybe it could be something that a player could request, but with a limit, like once every ten moves or so.  That way, you'd still be able to tell who's winning, but you wouldn't know exactly when your opponent made a blunder, so you wouldn't know if it would be something that you could take advantage of or something you could have taken advantage of a few moves ago.  It could also be something that both players would have to agree on.  Your opponent requests an analysis, and it would have a message in the game (like it does with a draw) that says, "[Opponent's name] has requested an analysis of this game.  Would you agree to the analysis?" or something like that. 

Then again, I guess post-analysis is still more reasonable than that.