Should torture be a punishment for horrendous crimes?
I would still support that the serial killer is publicly executed, so that other criminals are shown that justice is not to be messed with
The problem here is that the effect you seek is demonstrably weak at best and non-existent in most cases. There is no historical evidence at all that public execution, which let's not forget was at one time an event on a par with a modern pop concert, in particular, or executions in general have any deterrent effect. In fact, the very opposite was often the case. As the saying 'might as well be hanged for a sheep as a lamb' attests, criminals were prone to upping the ante committing more serious crimes on the basis that they were risking everything anyway so they might as well make it worth their while.
In any case, I have never understood the logic of death as the ultimate punishment, especially in an increasingly atheistic society where there is no question of hurrying someone on to face divine judgement (always a dubious theological justification in an event). Far from facing the perpetrator with their own guilt and remorse death simply releases them from any responsibility for reparation. It is those who are left behind who are being punished in reality and on the flimsiest of excuses, guilt by association. That is not justice, at least not by any sane, rational definition.