Is there a power that makes someone immortal?
Typically a recovery power or "living Dead" or some reskinned power thereof.
I think the power has to be at least equal in refresh to the power that makes them immortal. Often, though, this might simply be an aspect. So, I don't think it comes down to a refresh cost.
I think you'd need narrative justification and that's it.
Like a specific time of year.
It doesn't necessarily mean they have the creatures' catch. That is something completely separate.
How is said character able to do this, narratively?
1. I think it should be an aspect at the very least
2. It should require a FP to invoke said aspect.
3. They still need to be able to kill their opponent. This 'power' shouldn't give them access to any specific catch. So if the opponent is nigh invulnerable, good luck killing them in the first place
4. Meta-gaming, if you're sending this creature/person against an a PC, you should let them know that Character death is on the table.