The issue of capital punishment has all sorts of angles that need to be considered. There exists the possibility of executing an innocent person. But we could leave that issue aside for the moment. Then there's the issue of the purpose of executing. Is its purpose to act as a deterrent, or as an appropriate retribution or punishment instead?
Does it in fact deter? If it does should we execute people if the net sum of peoples lives are saved even if we regard capital punishment as barbaric? Or it retribution or punishment sufficient in itself even if it has no deterrent?
What about the cost of keeping them in prison for many years compared to execution?
What about the issue of the ultimate origins of our behaviour? Is it me who determines my behaviour? Or are we mere meat robots whose behaviour is determined by physical laws rather than our conscious intents? In which case why would any punishment be justified?
Who should we execute? Anyone who illegally and with premeditation murders someone else? What about a woman who murders her husband as a result of being beat up by him day after day, year after year? Or murdering anyone who has made your life a living hell? What about murder due to a hopeless all-consuming jealousy?
Is executing, in fact, a greater punishment than spending many years languishing in prison? That also depends on what will actually happen to that person after death.