I firmly support it, but I think it should be restricted to cases where you know without a doubt they did it, like when there were multiple disinterested witnesses. It should be done by hanging (as is traditional in America) as part of a voluntary public ritual.
I don’t view crime and punishment as REALLY being about punishment or rehabilitation, but rather about damage mitigation. When we jail people, it is neither about punishment nor rehabilitation, but a form of quarantine that keeps them away from us to minimize their harm. Execution is the ultimate form of quarantine since it permanently eliminates the source of harm. Public Hanging is a form of death which demonstrates the authority of the State while respecting the executed criminal and binding the community together in an act of lawful violence.
Execution would be the standard response to murders (but not a necessary crime), career criminals, treason, and the violently insane.