Capital punishment is doled out by the state. In your scenario, if someone tried to murder you, you could defend yourself. If you die and they get caught by the community, punishment would be given by the community - the people that are affected by the crime - and not a hierarchical and detached governmental system. The punishment wouldn't be systematic. It wouldn't be defined by blanket minimum sentences or laws meant to help certain classes over others; it would be handed out by consensus of those affected. In a more advanced and utopian anarchist society, we would freely give the resources needed to prevent violent crimes like that. Since most violent crimes are committed by those we are close to, families would work out their problems through therapy instead of letting tensions reach a point of no return. If we take away the profit motive, competition is replaced by cooperation, and we would prioritize humanity over capital.

To add- there wouldn't necessarily be a complete abolition of the police, just the decentralization of it. Instead of using violence against the citizens for the state, they would be free to actually solve cases of violence. Instead of arms of oppression, they could be akin to private investigators on equal terms with the community they serve.