Interesting... However, in the one song I was talking about (The Walrus), they say: I am he as you are he as you are me... rather than I am him as you are him as you are... Doesn't it look arbitrary?I am me and you are you sounds natural and colloquial.
Some people might insist that it should be I am I and you are you but it sounds stilted and pedantic to me.
The same thing occurs with It's me/I. Most people would say It's me. but the pedants would insist on It's I.
The reason is that the verb to be is a linking verb, and linking verbs join two words in the nominative case. In a fully declined language, like Latin, a sentence like You are happy would have both the pronoun you and the complement happy in the nominative case. But English has only vestigial traces of these distinctions, in the personal pronouns. Most people don't know much about them, and simply say what sounds natural to their ear.
Depends on context and intent:It's such an unnatural sentence that none of us would ever say it and actually know how to. 'I am I' is tempting for the obvious x = x sound, but normal English grammar makes 'I am me' more natural.
Sometimes we vary the natural cases in order to make a point. So 'I saw myself in a mirror' is the ordinary way of saying it, but you might say, 'I saw me in a mirror' to emphasize that you felt you were looking at almost a different person.