I've been entertaining what it's like to be immortal for quite some time. The state of unending, eternal existence appeals to a lot of people: The preservation of the purest forms of our minds and bodies, our most fundamental memories has a ring to it. In all certitude, people fear confronting their own mortality, the resultant suffering and the subsequent death. Immortality is the ultimate solution to that encounter, yet, I say this hesitantly, as death is truly something that can only be encapsulated face-to-face. The problem, however, quickly becomes evident. The state of immortality, no matter how wonderful, does not always equate to joy.
To have the purest mental faculties and enjoy their benefits in eternal, unblemished state is not the pivotal factor when it comes to whether immortality becomes appealing. What's more important is how those faculties manifest in such an eternal life. A constant state of perception will bring about a constant state of torment and pain. Regrets will begin to mount, pain will begin to feel unbearable, and immortality will eventually become a negative-sum game. But, when we realize that our minds will no longer take the form of the body we were born with, then it becomes more apparent why we cling to the body so desperately. We can't bear the idea of no longer taking up space, no longer occupying that body, even though it's a vehicle that will ultimately collapse. That idea can be articulated as a form of fear against the unknown.
What would drive life if it wasn't for death? Would we, as a collective, have achieved our milestones in the magnitude of the Industrial revolution, the age of science and the imminent technological revolution quicker if we were immortal? What about our very own bucket lists? The problem that eventually obviates itself is that, in being immortal, there is no ultimate deadline to anything. There is no incentive to keep running in a race with no finish line, for if we have the right to live eternally, what's the difference in our state of existence?
Ultimately, Buddhism sheds the most light in regards to Immortality. The cyclical nature of life and reincarnation it proposes inherently contradicts the concept of Immortality, where we do not die of any causes, be it natural, physical or by way of disease and illness. But there are many instances when people come upon some other, seemingly unlikely form of immortality. Many become monks, or nuns, or have some religious calling. Some devote their lives to a specific purpose, such as the creation of art, or science, or philosophy, etc. This is all well and good, except there is one problem. We do not exist. Our minds exist, but our bodies, the vehicle in which our mind resides, they do not. Therefore, we are nothing more than a mind floating in space. The body is not really who we are. This is the basic Buddhist principle that Buddhism does not claim to present a "true" doctrine of immortality. Rather, it presents us with a way of seeing our mortality, which is not as much a claim to permanence, but a realization that it's the mind that makes the body who we are. When our minds dissolve into space, what really matters is what our life brings to light, and the light our next life will bring. Perhaps not in the literal sense, but rather in living in people's memories.
Certainly, there are many accounts of the Buddhists and Immortality from their own scriptures. It is oft repeated that those who knows of the things past, present and future are those that have achieved a state of Immortality, if only a sense of it, that those that are wise and virtuous so as to know all these things should be ready to die and be reborn into a new life. The Buddha said that it is not possible for one to be Immortal, that it is a condition that one loses after death, if one has been a good or bad person. He said that, "A man may have thousands of sons, but still there will be many days when they cannot help him, as they are old and sick and do not know how to work. In that case he should be ready for death." So he said that, "A man can be immortal only if he is wise and virtuous."
Conclusion
In short, the concept of Immortality is not of this earth. There are many forms of Immortality, most of which are beyond this earthly experience. There are only two ways of becoming Immortal, one is by having a positive mind, the other is to become a wise man. Wisdom here is the capacity to perceive your place in the Universe. This is not about being an expert on all things nor about becoming a philosopher. This is simply an understanding of your place in the Universe. If you do not understand your place in the Universe, you will always strive for the physical, the monetary or the material and you will never understand, because you will always be living from a fear of death. All fear of death comes from a place of ignorance. Fear can come only from a lack of wisdom, and the wise man has conquered his fear of death and is now living a purpose-filled life.