Tyrell could be another alter ego. If we assume that Elliot's other alter ego represents some aspect of himself that he represses, Tyrell could be the same thing. Elliot's meetings with Tyrell tend toward the surreal which all of his hallucinations do, and in the last meeting between Elliot and Tyrell's wife they didn't have her act as if Elliot was a stranger to her. She acted really cagey when he asked if she'd seen him.

I don't think so. At one point, Tyrell meets with Elliot's boss.

I mean I doubt he killed Tyrell, but Tyrell did put on the threatening gloves in his apartment, so I'm guessing Elliot wanted to make sure that he walked out of the fsociety building alive. Makes sense he'd go for the gun.

Maybe he used the gun to threaten Tyrell after them finishing their work, and has locked him away somewhere. Maybe he shot to incapacitate him, and Tyrell is passed out in some hospital.

The voice on the phone at the end of s02e02 is Tyrell though. He's obviously on the run somewhere; most likely overseas.

We joked that Tyrell is another split personality, since the main character never really seems to interact with Tyrell and other people in the same conversations.

Tyrell's wife could know about his personalities and just not care, hence the weird conversation at the end of season 1.

Yes, but remember Elliot is a very unreliable narrator, and that's definitely a scene from his perspective.

I still think Tyrell is alive and was really on the phone, but the fan theory isn't nullified just because his voice was heard.