Marathon!Cortana, who is not quite the same character as the canon one; the basis for this version of her are the old “Cortana Letters” from ‘99, and all the theories it generated. I’d suggest reading this long-ass post of mine where I attempted to justify her presence on the Marathon when no one mentioned her.
The “abuse” mentioned above refers to Durandal’s selective memory-wiping as part of Bernard Strauss’s studies on AI Rampancy. Neither Durandal nor Cortana are happy to learn about this, for different reasons; they were previously about as close as non-Rampant AIs could be.
The latter’s connection to the former comes from all the old fan-insistence that, because the swords they were named after were said to be of the same make, Durandal and Cortana had to be directly connected. About half of Cortana’s code was copied over from Durandal’s, after which point the two developed independently.
Whereas Durandal and Leela use vessels mostly for utility and Tycho is locked into his as a sort of punishment, Cortana’s extensive vessel usage is a security measure--she does not want to get booted into another closed network.
MAY 2016 EDIT: And my fussy revisionism continues. This time I wanted to redo Cortana's avatar, as I was never really satisfied with just using her Halo 4 one wholesale. This one's based on some Halo 3 concept art and has a slightly more obvious nod to Durandal.
Also ended up redrawing all the side art, in part because my facial anatomy skills improved a good deal later on.