In English, the phrase alter ego means a second self or, more rarely, a very close or intimate friend. It comes directly from Latin, where it translates literally to other self or other I, and in English it retains its original form. It is two words, and it is not hyphenated.
Examples
Although alter ego is two words, some publications insert a hyphen to make it a compound noun—for example:
Like her Twilight alter-ego Bella Swan, Kristen Stewart should probably pay more attention on the road. [US Magazine]
But his alter-ego is Tom Trust, the aforementioned heel the little girl was taunting. [Myrtle Beach Sun News]
But most do not hyphenate alter ego—for example:
What emerges is an alter ego of London, essential to the city but apart from it, a magical-tawdry place … [Guardian]
It was Richard Muggles – Globe movie critic Rick Groen’s curmudgeonly alter ego – who reviewed the first film of the Harry Potter series. [Globe and Mail]
He even had a German alter ego named “Reinhard Pferdmann,” and shot elaborate war videos with his confederates. [The Atlantic]