Not wanting to die is fine, I think more importantly, not wanting to age and gradually become a burden towards end of life is also natural. Not wanting to be lost and missed (nevertheless hoping that you've done well enough to be missed). The idea of being bedridden and helpless is more scary to me than death.
But not wanting to die __ever__ can be driven by narcissism. If no one dies or gets old then we never make way for the next generation. If the cure is widespread you get overpopulation, if it is limited you get a class divide, never aging fantastically rich immortals and a semi-permanent underclass. Neither seems fantastic. What are the rules? Do you give something up to be in your 20s forever?
Generational change seems like a strong force for good in a lot of ways, to me at least. It can be harder to rewire a neural network than it is to start fresh.
Note that any non-magical intervention will likely be 'increasing longevity' not granting immortality. And even if that were the case, there's always the heat death of the universe. There's no escape from reality closing, you'd probably have a difficult time sustaining biological life-forms long before the universe is dark and cold. I doubt it would be pleasant. What is the storage limit for a brain anyway?
Personally I find some cold comfort that everything I do is meaningless and will be forgotten. I will once rest and be gone. So will everything.