sounds like shes getting really frustrated about something, two yr olds are often difficult they are finding themselves and realising they are there own person, hence they are often pushy at this age.
Could it be something she is eating, take note of things when she does go into one, it may be something really silly after all she is only 2.
My two yr old went through a strange obsession with spiders in the bath and this would set him off as he was sure we were hurting it, but being 2 they cannot get the words out and it makes them frustrated.
(I have 6 kids myself)
2 years old is pretty young to diagnose bipolar. The brain is still growing at that point. I'd talk to the pediatrician. If he thinks it warrants a closer look he'll refer you to a child psychologist or psychiatrist. It may be worth going to a psychologist yourself to help you deal with the situation. The more frustrated you get the more she'll pick up that emotion and mirror it back to you.
indigogirl's suggestion that you have an experienced parent watch you and your child interact is a good one. Two year olds go through what is called 'First Adolescence' in that second year, and it can be pretty difficult as the child tries first one thing and then another. Sometimes a parent, in a way trying TOO hard, unwittingly encourages unpleasant behavior in the child. But sometimes--as was the case with our son--the behavior comes from within the kiddo.
In either case, a visit with the pediatrician will be in order, because of course biting and hitting are not OK, regardless of age.
Do you have an older friend who has raised children successfully, who could come and watch you interact with your child? A child this out of control could simply need to be put in her room and told she can come out when she calms down. Is she taking any medications? My son reacted this way to baby sudafed, dimetapp, and other cold emdications (called an adverse reaction). i would definitely make an appointment with your pediatrician for help. make it clear to the child that she is not permitted to injure you, or she will be put in a time out.