If justice is merely a moral intuition, then it needs a democratic vote, and there is no barbarity or civilisation in the death penalty: only in rejecting the will of the people.
If we want consistent intuitions, then there's a discussion to be had to check whether people's opinions are consistent with each other.
If we want to have a practical argument without considering justice then the death penalty has little going for it. But considering an act of justice without the justice is utterly pointless. Furthermore, a lot of the cost of the death penalty is due to the endless array of appeals, which are hardly specific to the death penalty, but screw up a large number of areas of law, such as corporate law, in which companies quite happily plough money into pointless cases just to buy time.
I am not happy with 'an eye for an eye' because it is far too kind. If I am a rich merchant who owns two ships, and I have a rival in the area who owns one ship and I sink it, then sinking a ship of mine in return still leaves me with a monopoly.
My own opinion is that there needs to be no way in which someone can benefit from a crime, as well as there being an equal action done to that person in return. So in my example, a ship of mine needs to be given to the other merchant, and then a ship of mine sunk.
Murderers can't ressurrect their victims, and therefore cannot serve a full punishment. They can pay reparations to interested parties (including the state itself) and be killed themselves. If I think that it's fine for one person to murder another, then I can have no complaint when someone does that to me.