This was a tough one to figure out; what I do when I'm alone is pretty much what I do when I'm not alone. I'm consistent like that.
I do talk to myself, though. I recognize that I'm not the only one who does that (turns out it's actually quite common), but especially in the car, I'll have real conversations with myself in which I as the speaker am not me, but am the concept of what another, wiser person might be in the passenger seat. The real me remains silent, unless answering a direct question. "Not me"--the speaker--normally takes on the role of coach/teacher. "How do you think that went?" "What did you do well? What can you do better next time?" "Do you see what happened there? How do we avoid that?" Etc.
Also, if no one else is around at home, I will often sit down at the piano, lower my head as if asleep (sometimes curling down almost all the way to the keyboard), close my eyes, and enter a sort of "trance" state as I randomly hit keys that eventually become a coherent (sometimes epic) tune. It's very hard to wake up.
That's probably the weirdest thing I do.
On a related note, here's a quote from Dieter F. Uchtdorf:
Pause for a moment and check where your own heart and thoughts are. Are you focused on the things that matter most? How you spend your quiet time may provide a valuable clue.
(You can read the rest of his talk here; it's one of my favorites.)