There are various retributive-justice arguments in favor of capital punishment for especially heinous crimes, none of which require that the crime be murder or leave permanent unmanageable suffering. Certain crimes shock the conscience. If the criminal raped and tortured a child, for example, a lot of people will think that execution is the only appropriate punishment regardless if the child survives will eventually recover.
First of all, there is the perceived-justice angle. Will the victim, the victim's family, and others sympathetic to the victim believe that justice has not been done if the perpetrator continues to live? If so, then there is a cost to society in not executing the perpetrator in that it will tend make people cynical about justice. "They care more about the criminal than the victim", etc. Over time, this cynicism about justice may lead to a less law-abiding population.
Second, there is the vengeance angle. If you don't punish the perpetrator sufficient to sooth the anger of the victim's family, it is more likely to create friction between the family of the victim and that of the perpetrator. Someone in the victim's family is likely to try to take vengeance on the perpetrator or someone in the perpetrator's family. Those of us raised in Western societies and not in dangerous neighborhoods often don't see the risk here because we are used to letting the government handle everything, but people who grew up in dangerous neighborhoods or non-Western societies, are far more likely to have experience taking the law into their own hands and be willing to do so. The way we broke that habit in the Western world is in part through the use of capital punishment.
Another consideration is abstract justice. If you believe that good and evil are real things, that there are real obligations and real obligatory justice, then you believe that there is simply a proper response to a heinous crime. The universe has been put out of balance by the crime and to put it back in balance, the perpetrator must lose his life. I'll also note that if you don't believe that good and evil are real, you shouldn't be concerned about capital punishment except for it's functional use. If evil isn't real, then it can't be evil to execute someone.