The first episode featured Naomi's battle with D'anclaude. When he stabbed her and then she electrocuted him? That was cool.
That whole battle, really. Dramatic speech atop the Martian city heights where we suddenly learn that Armitage can control machines with her mind? Running directly at D'anclaude's bullets and a grenade without stopping or caring for the damage?
Also in the second episode when Armitage confronted two lecherous punks in a dark allyway, punched one in the groin, and then brutally interrogated the other, scaring the living crap out of him.
Episode 3: Armitage self-destructs to get rid of the mecha-D'anclaude. She doesn't die, but she does seriously scramble her CPU in the process.
And in episode 4 when Armitage shout that she will keep fighting to stay alive and then the reprogrammed D'anclaude referring to her as "refined".
Later in that episode, Armitage and Ross really do start fighting during their last stand against the military. Jet-powered wings, arm-laser cannons, and Ross in a mecha suit: it's not what you expect, but it's definitely awesome.
There's a cool scene in Poly-Matrix that bridges Ross's arrival on Mars and his first meeting with Armitage. In the interim, Armitage apprehends some (cyborg) punks by totaling a bike into their faces, bisecting a guy's head with the wheel, and then declares to her bemused colleagues that she won't be paying for the damage. Badass.
Here's the early part of the film when Armatage dukes it out with a man in a large robo suit. She makes a large jump attack on him with a large metal rod, and gives him a nasty suplex!
When Ross was just a second away from getting gunned down by a helicopter, Armitage distract it, then shoots the pilot in the face!
As she rides the elevator to confront Dimitrio, she looks into the camera in the elevator saying, "If you harm so much as a hair on my daughter... I will kill you!"
The entire Final Battle, in which Ross and Naomi fight tooth and nail against a pair of seemingly indestructible Evil Knockoffs of Naomi, all for the sake of protecting their daughter Yoko. Also falls into Heartwarming.