Is it possible to have version control on the about me section (or any content for that matter) in the personal profile?
The About Me page on the My Site Host site pulls content from the User Profile Service application via web parts and controls. The only user profile content available (name, phone, etc.) is the latest information -- user profiles don't have versioning.
However, if any content that is displayed via pages on the My Site Host reside in lists/libraries on the My Site Host then yes, those lists/libraries can have versioning enabled. The same is true for content residing in lists/libraries on a user's personal My Site site collection.