People talk about immortality like they know what it is. They say it is "living forever". But we can imagine somebody who experiences life getting asymptotically slower and slower, so that they never experience their own death (or perhaps experience it "after omega"!). But somebody else would see the person die, and declare that they were not immortal. Likewise, if someone was immortal in the normal sense, but experienced life getting faster and faster, they could "burn through" an "infinite" amount of time. What I am getting at is that time in isolation might just be a topological interval, and that it is our minds that institute a distance on it, evolutionarily springing from the other laws of physics. So, is immortality a property relative to experience?
Well I was thinking about this question for a while and here's what I came up with:
The main problem is that you define immortality using time ("living forever"), and time is relative in its very nature, whether we look at it as a physical property or from a more common point of view. So what causes immortality to appear as relative in your example, is the fact that it's defined using time. On a side note, I don't think you can define an absolute value, when you base your definition on a relative one.
Let's see what happens, if we define immortality as "not being able to die", a definition that also comes natural. It implies, that if anyone can see you die, you're not immortal, but in fact, dead. This fixes the black hole paradox, because living forever doesn't equal immortality in the above definition, as long as you can be seen dead by any observer in any circumstances.
Translating it to a more mathematical definition: a person P is immortal if and only if for each observer O in time and place P is not seen as dead by O. In other words: a person P is not immortal if and only if there exists an observer O that can see person P as dead.
My hypothesis is that the 2nd definition describes the concept of immorality that can be found in the culture better. And although in most religions it is said that afterlife is "living forever", it is also said that souls cannot be killed or destroyed, and that is what grants the afterlife immortality. And more than that: spirits and souls are not bound by any laws of physics, including time (at least in Christianity, I'm not a religion expert). Also, note that you don't even have to be seen as alive to be immortal in the 2nd definition.
So, assuming my hypothesis is correct, immortality is not a relative property.
By the way, here's another paradox: is the Shrodinger cat immortal? It lives forever, if you never open the box (immortal in the 1st definition). But it's dead at the same time, so it's not immortal in the 2nd definition.