The Crow/WMG

Albrecht was formerly a cop in Gotham City, or some other city home to superhumans.
Explains how easily he goes along with the idea of a guy coming back from the dead.
 * Also, in Tim Burton's Batman, Commissioner Gordon addresses a young black cop as "Albrecht".

All the supernatural stuff is just an hallucination.
In the comic, at least, the circumstances of Eric's death and ressurrection are a lot more ambiguous. It's possible he was only clinically dead for a short period. His visions of the afterlife and apparent inability to feel pain are simply due to one or more factors including brain damage, drugs and/or simply going insane with grief and anger.

The various Karas warriors are incarnations--or even a sort of "sub-species" of The Crow.
"Karas" actually means "raven" in Japanese, and the supernatural elements, including the near-death nature of the protagonist, seem to match up. The fact that every city has a Karas may hint at a different but related type of power.

The Lightning gave Eric a Power Boost
Shooting the Crow has made Eric vulnerable to knives and guns among most things, by all intents and purposes the lightning that struck the weather vane he used as a makeshift sword against Top Dollar; should have killed Eric instantly when he touched it. Instead he screamed in agony for a moment and then pulled the vane out to fight. This troper believes that the lightning and whatever force that caused the crow to bring Eric back, gave him a temporary power boost to give him a fighting chance before he could pass on.