Eddie Carbone's complexity grows out of his lack of ability to see himself clearly. Eddie has developed a romantic love for his niece.
Eddie has unarticulated and perhaps unconscious desires to possess Catherine himself.
He does not fully recognize his feelings for what they are. Instead, Eddie insists that he is protective of Catherine because that is his role. He has raised her as a father. He refuses to admit that his love for Catherine is something other than fatherly love.
Eddie's mercurial passions stem from the conflict created by his inappropriate feelings for his niece. He cannot control these feelings. The fact that he is incapable of admitting to these feelings creates much of the complexity in his character.
Additionally, Eddie exists in a cultural situation that limits his ability to take revenge against his rival, Rodolpho. The one course of action open to him is reprehensible. Eddie cannot, morally, turn in his cousins to the immigration office, though it is his only legal option if he is to avoid losing Catherine to Rodolpho.
To report the two to the immigration office may be the legal recourse, but, as Eddie knows, it is not the moral choice.
This lack of options but need to act leads to another complexity in Eddie. He is generally a "good man", but he is carried away by his passions and his weaker impulses. He violates the code of his community.