"Mein anderes Ich."
I am unsure how this phrase would be used in practice, even in English. Anyone care to give examples in either language?
Well, in English you could say something a bit introspective like, "I am two people. One self enjoys X, but my other self enjoys Y." It'd be a bit weird but it could happen.
Perhaps when describing something that seems contrary to other's perception of your personality. Like if a literature professor told you about his/her weekend boxing matches, maybe. Just a guess.
In English, it is: 'My other self'. Which is more of a slang or Colloquialism.