Naked mole rats are social animals that live in colonies. There are differences of course, but I do not think the evidence suggests that aging is a 'requirement' for complexity. The simplest explanation is that aging is an evolutionary adaption that gives species that age an advantage. Human cells obviously possess the potential to persist past their programmed deaths - that's what cancer is. So I think it's obvious there are no biological limits to us stopping our physical age at whatever stage we want - but I'm equally sure that would involve changes to the experience of being human we may not like (perhaps being in the body of a 20 year old means being in the mind of a 20 year old, for instance).

They seem to have a biological clock and are just more isolated from the effects of aging. Humans are similar that we live three to four times as long as other great apes.

The best example of human cells having programmed death is simply the main exception: offspring.

Children are made from a cell of the mother, combined with some DNA of the father. It is not created out of nothing or from scratch, it quite literally just one of the cells of the mother. So if a child survives one of the cells of the mother survives the mother's death.

Obviously the process involves resetting the "death counter". In fact there's a disease where that step seems to have gotten skipped: progeria.