I think immortality means impervious to death. That doesn't mean you can't get badly injured or die but it does mean that you'd resurrect good as new by the next day. Kenny on South Park and Bill Murray's character in Groundhog Day were immortal.

@Sgt_Spanky I had considered a few types of immortality. But he said that you "LOSE" a limb, whether he meant to or not, the implication is that you are not refreshed in any way. You would have to live with that lost arm. You are ageless, but vulnerable. This immortality has its cake and eats it, too. You get to live beyond your cohorts and decide when you want to tap out.
However, if we were to consider your interpretation, the only downside would be the pain to getting an arm cut or an infection. It would be a real trip to exist beyond the end of the universe, albeit boring for a couple trillion years. Not to mention the suffering of floating through the vacuum of space, dying only to be revived moments later. Or getting caught in a black hole and not be able to respawn anywhere safe after that. Actually, that would be the best thing that could happen to you because time would go by much faster inside a black hole. A year in there would probably be a billion out here.
I'm not sure what the rules of this immortality are according to the scientist guy or even the OP. I would probably consider not taking the second kind of immortality.