Plenty of fictional characters have sought immortality, but most of them are villains. Perhaps that’s a sign?
The benefits of never departing this mortal coil are self-evident — none of us knows for certain what awaits us, and being alive is generally pretty cool — whereas the downsides take more sussing out. You’d lose everyone you know and love, and living through so much change could embitter you to the ways of an ever-changing society. Also, forever is a really long time.