I haven't seen the Animatrix, but there are multiple dimensions in CoH, with some being more deviations than others. Your character could have developed their powers and, in the process of doing so, hopped dimensions. Just remember that what might have held true in your character's original dimension, might not hold true in the Primal dimension.
Here is an example. My character Hawtdog is from a rather silly dimension, and it colors her experiences and expectations. She firmly believes she was sent into the past by her Leader to prevent humanity from being backstabbed by Catgirls and, ultimately, enslaved by aliens. In reality she was shunted from one dimension to another where the event she was supposed to prevent has never and never will occur. Most of the historical information she knows (which, admittedly, isn't much) is worthless. Her understanding of certain things, like magic, comes off as flawed. She's just a little special so she doesn't let this bother her, or at least doesn't dwell upon it.
So the questions I guess you'd need to answer are.
How did your character get their powers?
What was the event that caused your character to hop dimensions?
Is there someone chasing them? If so, Who are they? Why are they after them? How is your character going to elude them?
Is returning to their home dimension something your character wants to do?
If you want to have an organization from their home dimension popping up here is a suggestion, keep them largely limited to their home dimension. They send operatives out to other dimensions to do very specific missions (like recapture a wayward speedster), but they dont have any major bases in other dimensions because of their potential discovery by other groups who are aware of the multi-dimensional nature of the multiverse could cause all sorts of problems for them.