First of all, I believe he could be a spirit, moreover, a ghoul. The definition of a ghoul in the dictionary is ‘a spirit interested in morbid things’. The fact his name is Inspector Goole, a homophone of ghoul, provides more evidence that he could be a ghoul. The fact he keeps bringing up the horrible images of ‘burnt her inside out, of course’ and ‘she died in great agony’ prove that he is very morbid.
I think it has definitely been proved that he is a time traveller. He knew that Eva Smith would drink disinfectant even before she had drunk it. The phone call at the end of the play proves this is the case, e.g.” A girl has just died, on the way to the infirmary, after swallowing some disinfectant.” Plus, when he came into it, it seemed that he already knew everybody’s involvement in the case, he just wanted them to admit this for themselves. For example, Shelia says “why-you fool-he knows...”. Also, he may possess magical powers, because in the film adaptation, they trap him in a room, then when they go to confront him, he has disappeared.
There is one thing we know about him for sure, however. We know that he is definitely not a police inspector, as Gerald asks a police sergeant and asks about the inspector, the sergeant swears there isn’t an inspector Goole on the force, e.g. “He swore there wasn’t any Inspector Goole...” This moreover proves he is definitely a liar.
Also, I believe there is a possibility that the inspector is the ghost of Eva Smith, or perhaps, her unborn child. I think this because the inspector knows every detail about Eva, much more detail than you would get from glancing at a diary. Plus he seems to know the feelings that she had deep down within herself, which you would only know if you were the person...