As a first approximation to the true ontology of personhood, I'm going to talk about three degees of persistence. First, persistence of your current stream of consciousness. Second, continuous existence in time of your physical or ontological self even through interruptions to consciousness. Finally, creation of a new self which is a duplicate or approximation to an earlier self which was destroyed.

I consider it very likely that the self - whatever that is - exists continuously throughout any interval of uninterrupted consciousness. I consider it unlikely but remotely possible that sleep, and total unconsciousness in other forms, is death to this self, and that each time we awake we are literally a new conscious being, fooled into thinking that the experiences of its predecessors also happened to it because it has their memories. I consider it far more likely that the self - whatever that is - lasts a lifetime, and corresponds to some form of brain activity which persists continuously even in deep sleep or coma, but which will certainly be terminated or at least interrupted by something as radical as cryopreservation. I consider it unlikely but remotely possible that some form of reanimation from current techniques of cryopreservation constitutes second-tier persistence - an interruption to consciousness but not an interruption to basic identity - but I think it far more likely that reanimation from such a condition will require the creation of something that by physical criteria is a new self.

I'm not totally against seeking third-tier persistence but that's not what I would mean by immortalism.