Obviously this will be subjective, but…
Mozart died before completing the Requiem and it was finished off by some people his wife hired. According to the Wikipedia, since then a few different people have tried to finish the piece(s) off in ways that might be more in the master’s style. But of course there’s no guarantee that any of them were particularly closer, that they were better composers than the people his wife hired, and I don’t know if any of these have had a high quality performance by professional orchestra.
So, has anyone heard and recommend any particular alternate versions? Or is there really no advantage to trying any of these compared to what is the “accepted” version?
If memory serves, Mozart had most of the framework of the requiem drafted out. He didn’t just keel halfway through the Lacrymosa, as the legend often goes, leaving poor ol’ Franz Xaver (Süßmayr) to finish the rest. If anyone could finish it for him, it was Süßmayr – sure, there are arguments about whether or not it’s truly in Mozart’s style… but seriously, the dude had been Mozart’s copyist and close friend for long enough, and Constanze was happy with it.
More often than not, this is the accepted “urtext” of the Requiem, and therefore it’s the one most often recorded and performed.
An interesting version of the requiem is the one recorded by John Eliot Gardiner, using a boychoir and men’s chorus, with period instruments. It’s not what you usually hear, that’s for certain… some consider it one of the closest, true-to-ol’-Mozart recordings out there. Me, I’m fond of the over-dramatic Neville Mariner approach to the Requiem. 
Hope this helps a little.