I used to do that all the time back in the 1E/2E days of D&D. I would DM for my buddy, and every so often he'd switch and DM for me (though only very rarely did the characters meet). It was a blast (literally, since both our solos were wizards), because with only one PC we were able to deck them out with crazy stuff and not have to worry too much about balance. They'd tromp around the Realms (the Forgotten ones) and do whatever they wanted, depending on what their goal was at the time. The adventures were largely driven by the desires of the PC, with very little prep on the DM's part. Our characters were "sort of evil" too, which was pretty fun at the time.
Hell, we still fondly mention those characters 20 years later. I think its just as possible to do in 4E now (and 3.x, which I don't care for and never invested time/money/energy in). Hopefully, I'll get the opportunity to try it some day.