Many of the self publishers I know write under one or more pen names. I occasionally get questions about the legality of this, whether or not it interferes with copyright. It doesn't matter to the copyright office if you want to use a pen name, they even make special provision for it, but they do advise that you don't use the pen name as the claimant. Having a fictional identity as the claimant may lead to difficulties if you need to defend the copyright from infringement, unless you're only interested in make-believe damages.
But pen names can be self defeating for self publishers, who have usually base their reputation on their name, rather than the name of their publishing company. I sometimes advise self publishers not to use their own name for their publishing company just to get past the snob hurdle of, "Oh, you published your own books", and it would certainly strike me as a little backwards to name your press after yourself and then to write under a pen name. But enough self publishers use pen names that my little lecture on the subject turned into my longest video yet:
Reputation and self promotion are so tied up in selling books that a pen name can only be useful if it goes along with a real person who can meet and greet, at least virtually. I'm always amazed when I hear from self publishers who think that anonymity is both a goal and a necessary ingredient for success. You aren't "getting away with something" by self publishing your books, and the only way you'll remain anonymous is if you fail. If nobody has ever heard of your books then they'll never have heard of you or your pen name and your anonymity will be safe. Doesn't strike me as much of a goal.
Promoting your books doesn't have to be a full time job, but it's not a job you can hide from if you want to make a living. It's not just about standing on a chair and yelling "listen to me", you have to convince people that you're somebody worth listening to, or reading. Your name is how people identify "that author I like" when they are shopping for fiction, or "the author who knew his stuff" when they are shopping for nonfiction. If you go the multiple pen name route, you are dividing your credibility over a number of reputations, just as if you put a bunch of websites online and divided up your PageRank. So before you rush into using a pen name without needing to, remember the famous words of that guy in the video. "You can fool illiterate people with a pen name, but they don't buy books."