By display name, do you mean a different username? Or a different blog name (such as blog1.wordpress.com and blog2.wordpress.com).
If you are trying to keep your two lives (work and personal) completely separate, then you might want to consider having two usernames. While they aren't visible often, when you make comments on others' blogs (and sometimes on your own), your username is displayed and so you may want to consider separating them.
If that is not as much of a concern, you can keep just one account and create multiple blogs which are totally separate. There is no way to tell that they are all run by the same person unless you draw that connection. You can create the additional blogs by following this help article: