I don't think it is immortality but a fear that their life may not have meant anything and they want to be remembered. It is sad to see people who have no one to remember them or what they have done, which in itself is a form of death.
I do not fear death nor what happens to the body.
Part of what makes the human race so interesting is our amazing amount of attention to life and death. The search for immortality evolves along with us, and shifts with every generation. In ancient history it was a search for trees of life and pools of magic water, and these days it is documenting our legacies and preserving our data. This obsession with youth and legacy is ingrained within us, and what drives us to explore, create, and invent. We are searching for life and time -- the very things we cannot fully ever comprehend. But this is what makes us human, and as long as we embrace it, we will continue to be such fascinating creatures in the world.
No, in fact, in the United States it does not seem as though we care much about morality at all any more. Morality is not as important as it used to be. In Middle Eastern countries they may stress morality and moral law, but the United States seems to take the opposite stance.
I think that in today's society we are keenly aware of our mortality. Especially in a post-9/11 world and with social media sharing so much of people's deaths, the fact that life is short is too obvious to miss. While we like to remain youthful for sure, I think a part of that is the dream to enjoy every bit of life, knowing full well that it will not last forever.