I don't even remember him killing her if he did. *EDIT* I remember now. I was thinking of his social worker. I thought that was just part of him trying to escape. I remember her telling him he's about to be SOL, and that nobody cares about them. But even with that I see him having anger directed at certain people, and pushing back against a system that's throwing him to the side.
The killing at the end to me was still him having anger directed at one person that wronged and humiliated him. He didn't shoot anyone else in the studio (like the old lady that was also a guest). In the cop car when the cop says look what you did as the city burns, and he smiles and says "ya" I could see a hint of him loving chaos.
Just to me he never went full psycho. He just didn't mind killing people that disrespected him.
That's actually what I LOVED about the movie. We seen joker causing havoc in a few movies. I didn't need another movie of just that. I loved that they explain HOW he became a psycho murderer. I also noticed that he became more unstable as he lost more support n connections. Work, therapist, imaginary gf, mom, career