I would like to point out that "bipolar" as a word is an adjective. Therefore, I am bipolar. It's the same as saying am tall. See, "I" is the subject of your sentence. "Am" is a linking verb which connects the subject with some form of adjective. (Adjectives are describing words for those who don't know their grammar.) "Bipolar" is the adjective linked to the subject.

To complete the picture, I should bring up "disorder" which is a noun. With that it is correct to say, "I have bipolar disorder." In this case "bipolar" is again an adjective, but here describes "disorder".

By itself, "I have bipolar" is grammatically incorrect.