It's interesting and I've never seen it before. That eye sound makes me think of
Levi and
Eli (with their American pronunciations) so I'm guessing probably American, possibly Jewish, possibly some sort of fundamentalist
Christian... *looks it up on wikipedia* and possibly a family name: the first bearer listed was named after another Amzi, and unusual names often run in families.
I don't like names with the eye ending much, so I'd prefer it as am-tzee.