Subverted: Jack appears to be killing crooks left and right, but it turns out he didn't have control of his actions at the time; when he's back to normal, he's The Cape, through and through.
Double Subverted: Jack is revealed to have been Brainwashed and Crazy, but underneath that...he's just as brutal.
Parodied: Jack tries to bring about his own brand of harsh justice, but he lives in a Sugar Bowl society where there's hardly ever even any conflict — Billy spends most of his time angsting that there's nothing to Angst about.
Alternatively: Jack is arrested for killing a criminal because, as it turns out, murder is murder.
Jack kills a "criminal", only to find out the real villain's still out there. This leads Jack to question his methods, feel guilty, and wonder if all those lives he's taken away before were really the lives of innocent men & women.
Jack kills a murderer, only to find out the killer was another Anti-hero who only killed villains like him.
Jack seeks therapy to help him become a more mentally stable individual.
Billy decides to be even more careful not to kill an innocent (wo)man again & take time to make sure he has solid evidence against them and that the evidencewasn'tplanted there against the suspect before killing them.
Billy tries to learn about the villain's motives & make sure the villain(s) isn't really another anti-hero before killing him/her/them/it.
Plotted A Good Waste: Billy's morally questionable actions makes him a more complex character and are used to analyze what "the qualities of a hero" truly are, and what it is that makes someone "heroic".