thinking about how Azula emphasizes togetherness in a way that no one else in her life really does, ie.
"I'm going to need a small elite team" (return to Omashu)
"I've plotted out every move of this day, and the only way we win is together" (crossroads of destiny)
"But I thought we were going to do this together" (Sozin's comet)
All of this compounds back to, while she was of course impacted by her mother's view of her which is something she can acknowledge ("my own mother thought I was a monster" / "I know what you really think of me. You think I'm a monster"), the implicit trauma Azula never acknowledges is that she doesn't want people to leave her. She relies on fear to control people so they don't betray her, but to her that has often been synonymous with leaving. It also speaks to some awareness about Ozai's true nature—since Azula could believe Ursa left entirely of her own accord, and Mai and Ty Lee did—as Ozai made Zuko leave. "You can't treat me like this, you can't treat me like Zuko!" is about disgrace, distance, and punishment yes, but is also about separation ("I deserve to be there by your side") and her fear of it... especially because as of this point, Zuko likewise left the family, meaning that Ozai is all she has left. It also shows in Azula's paranoia, with the punishment she doles out to people being banishment—that they have to leave because she says so, that she's self-sabotaging and making them leave before they choose to first ("sooner or later, they all would've betrayed me, just like Mai and Ty Lee did").
This fear of abandonment makes me think most notably of Katara and Aang. Partially because Katara shares this fear, but is also incredibly steadfast. While she would likely hold a lot of resentment and hatred towards Azula post war (the girl did kill Aang) and I don't know if their personalities would mesh well, Katara's loyalty ("if he's banished, then I'm banished too!" / "And we need you" / "I will never ever turn my back on people who need me") and friendship (or more) could be a balm to Azula's abandonment issues. Alternatively, Aang is someone who like Katara — if not more so in some ways — values togetherness and presence ("we've got to stick together" / "we could learn it together!" / "but we just got the family back together"). Being abandoned is something he fears (keeping Hakoda's message) but ultimately something Aang is more willing to undergo and handles a bit better. At the same time, his added maturity here is juxtaposed with his immaturity and tendency to leave / run away, something that Katara doesn't really do except when it comes to certain things.
Basically: Azula has a deep seated fear of abandonment, and I think it makes Katara and/or Aang very interesting potential dynamic partners for her either way.
