That would make an interesting story plot. Suppose we've developed the ability to copy a consciousness. It has all your memories, all your feelings, your same sense of "self" or identity. If you die, you experience death, but the copy of your consciousness lives on, as a perfect replacement. Would that be immortality?

I have thought about this quite a lot

I don't think it is immortality. It is just cloning

Any theoretical scheme that could let you exist at the same time as a clone of yourself means the clone is clearly not you. It's a different independent individual that only appears to be you

Altered Carbon, Richard Morgan 2002. There's also a Netflix series.