"Compensation" would not be achieved, of course, because the victim doesn't actually receive anything of material value, but well targeted "Retribution" is a useful tool for society.
It's about instilling fear in future would-be murderers so they never kill in the first place. For that, you have to set up an example. Waiting in prison for an execution date is at once humiliating and demoralizing, because you know its inevitable. You can do nothing to escape. A future would be murderer experiences that pain by way of imagination of how it would feel if it happened to them. Not everyone is motivated the same by a fear of being in that situation, but many people are. Some would be more afraid of a life sentence.
So the answer is: do both. Whenever a prosecutor perceives that life in prison would be more painful to the defendant, they should ask for life. If they perceive that execution would cause more distress, then they should seek the death penalty. It never hurts to have more tools in your tool box. Some problems are best addressed with a screwdriver. Others require a wrench.
I think a barbaric solution is better than no solution at all. Some people don't respond to civilized approaches.