This is absolutely outrageous.
We are already immortal. My children and my children's children are my continuation. They are me. More than that, your infinite existence is harmful and is a threat to your children, who are essentially you. It's very important that you die. Otherwise, your children will have to compete for resources not only with everyone else, but you as well.
On top of all of that, we all go through phases as we age. Those phases are essential to our maturing and experiencing life. To stop aging is the equivalent of stopping the maturation process and stunting life itself.
Death is as natural as birth. We are born, we grow, we experience, we die. To break that cycle is a perversion in itself and would destroy any concept of humanity that currently exists.
We should fear untimely death, certainly. However, timely death should be embraced as much as birth. It is the completion of a cycle. There is absolutely no need to fear it.