I believe your friend was incorrect, in general. In my experience "I'm taking off" is a perfectly natural and common way for people to say that they are leaving. As this is a colloquialism, however, it is quite possible that it is more or less common in different places or amongst different groups of people, so it is possible that amongst the group of people your friend knows it is not as common, and thus sounds more unusual to them.
In my experience, "I'm gonna take off" and "I'm taking off" both have roughly the same meaning. They are often also said with "now" on the end, but whether or not they have "now" they all have an implication that you are actually leaving (i.e. just about to go out the door) right now.