Proceed with caution and don't rush into it - if she truly cares about you, she will accept that the relationship needs to deepen and progress at a pace acceptable to both of you...not just her personal timetable.

Maybe her mood is towards a manic state, maybe not. She may have a clingy personality that has nothing to do with bipolar disorder. I have bipolar disorder and I have never jumped into a relationship in this way - we are all different, though, bipolar or not.

My impression is that she sounds like a bit of a "handful" in a not good way - but if you are into her, give it a shot but don't let her control the pace of the relationship. A relationship takes two and it's not fair to either of you, in the long run, if she is diving in head first and you aren't sure or "ready."

Best of luck.