Some movies require someone to watch them with. Others are much better and more gripping if you are watching them alone.
All is Lost
This movie is about Robert Redford, out on the open seas, when he gets into a spot of trouble. The almost complete lack of dialogue and Redford having to save himself when trouble hits will grip you more if you don't have someone to lean on.
The Shawshank Redemption
No one ever wants to be put in solitary but you'll want to be nice and solitary when you watch one of the best prison movies ever made.
Into the Wild
This is the true story of college graduate Christopher McCandless who wandered off into the wild. Sadly he died a few weeks later. I recommend watching these dramatic events unfold alone as he was.
Lost in Translation
While this movie isn't strictly about being alone, it is about two people who thought they were. You'll be able to feel that atmosphere if you watch it alone.
No Country for Old Men
This is one of those movies you need to be able to pay complete attention to from start to finish. Having someone sitting next to you might distract you. Turn the lights off, pull the shades and watch this all alone.