(spoilers) Bloody rags
The first time he goes to her apartment, he sees the bloody rags, which could not have happened in that timeline as it did. The one condition for his bloody rags being in Claire's apartment is that she is alive and they went back to her apartment so she could clean him up. It doesn't make any sense that his blood is in her apartment in the timeline where Claire is dead - because he was never there.
However, later, when he does actually go back in time, the movie is trying to convey the time travel theory of "even if you go back and try to change time, it will have turned out that you actually already did that, and everything is as it is now precisely *because* you went back -- and nothing has changed, and everything is as it will be, forever, as everything has sort of 'already happened' and/or is pre-determined like a recording". They try to push that idea when they go back to her apartment (when Claire is alive), and he realizes, "Oh, oops, here's my bloody rags, I remember this - I haven't changed anything at all!"
That's fine - that means Claire is ultimately going to die again, and the ferry is going to blow up and everything will happen as it already did - I guess. But then it doesn't. Why is that? Because he *just really wanted* it to happen differently this time?
I know it would've been a crappy ending if she ends up dying anyways (and his clone would have to disintegrate, too..), but I couldn't get over the "the bloody rags still exist in the timeline where Claire isn't alive" thing.