Fighting death is more about the immortality of the body. The mind is recycled much sooner. The "I" is created by the mind anew every moment. Even sleep is like a mini-death, because we "don't exist" for a short time, then we continue existing again.
But, of course, if we actually get to cheat death and live forever, a new kind of life opens up. If we can back up our minds and then upload them into a young body or in a virtual reality, then we can also be more like a character in a game. We could have many "lives", we could have "save checkpoints", we could restore from previous points, or fork and create more of us. In fact life would not be the same as it was during "linear life". Would a meta-life such as this be the same as immortality? I don't think it would. It would be a completely different thing altogether.