Is being brainwashed or hypnotized comparable to being in a coma? One has no control over the brain right? And the way the brain works is radically affected. Right?

Look up Democratic party and it describes just as you stated. 

No.  When you are comatose the brain is compromised through injury.  When you are brainwashed, you are more like hypnotized.  

People come out of both "trance states" quite often.   Some do not.   Hypnotism is merely a suggestion.  So is brain washing.  When you connect this action with a skilled hypnotist, or skilled manipulator, in the case of brainwashing, patterns begin to emerge that the brain gloms onto for different reasons.   With hypnosis, it is usually a seduction that makes one form the "new belief".    For instance, if you want to quit smoking, the hypnotist recites the desire to be slim and beautiful and not feel hunger.    In brainwashing, it is usually caused by the part of the brain that is hell-bent on survival.  Deep within our brains, we are programmed to survive at almost all cost.   When faced with this life-or-death situation, the brain will accept and adopt whatever it is that their captor leads them to believe.

Now a coma is a different state.  Being on the  outside looking at the comatose patient, most of us  think he is out of reach.  Out of touch.   When awaking from a coma, I believe, most people have no memory of where they were.  There is no controlling anyone at all when they are totally unconscious like that.