Spell:
Antimagic Field
Abilities:
Mummy Lord, Lich, Ghost, Mythic Hero Immortality
Several creatures and even PCs can have abilities that allow them to return from death/destruction after a set time unless certain criteria are met for their permanent rest. This set time ranges from 24 hours to 2d4 days or 1d10 days. These abilities are all supernatural, however, meaning an Antimagic Field suppresses them.
1) If the Antimagic field is temporary, does that mean the creature's respawn countdown is on pause while their ability is suppressed, or are they counted as not having the ability at the time of their destruction and become permanently dead? For instance, a normal PC once slain is dead and the spirit goes off or whatever. But a Mythic PC has Immortality. If that Immortality is suppressed and they are slain, are they counted as if they never had it?
2) If the Antimagic is permanent, like being on a Plane with the Dead Magic trait, does that mean a Mythic character who dies there, is dead and their spirit moves on? Does retrieving the body change that?
Bonus) If there is a portal to a Dead Magic plane, as made through the Create Greater Demiplane spell, and a Ghost is forced through the Portal, what happens to them? I would expect and an incorporeal they would wink out, but since it is a permanent effect, are they permanently winked out? Could any magic reach them or affect the Demiplane with that trait? Can you disjoin a Dead Magic plane?
For the non-mythic abilities, I would say that the "rejuvenation effect" is indeed suppressed by the AMF.
It would take a mythic AMF to do that to the. mythic immortality ability.
However, the ability is suppressed only while in the field. It is never removed. The one with the ability is only dead as long as the field persists. Removing the body from the field restarts the clock right where it stopped.
Yes, you can disjoin a dead magic Demi-plane.
Hard to say whether antimagic field matters to Immortal (Su). Is it an ability of the body or of the soul? If the latter, where is the soul in the interim, and can its abilities be affected by an antimagic field on the Material Plane?
Continuing my answer:
2) Permanent or not, the anti-magic field cannot dispel supernatural abilities, they can only be suppressed. Suppression means it is still there, just isn't fully functional. As a result, the creature would "die" and the soul would pass on, but if the body is ever removed from the effect, the ability would restart and they would be brought back to life just as normal.
Bonus) I'm not positive that you can "force" a incorporeal creature through a inter-planar gate -- that would definitely have to be some exceptional circumstances.
A "Permanent" effect is more accurately termed a "persistent" effect -- it can be affected by dispel magic/disjunction, and once that goes away, the ghost would re-appear, just like the the rejuvenation ability would re-start and apply as normal.