Hm... Is it technically possible -- however fantastically, ludicrously unlikely -- that quantum teleportation could transport you from inside to outside the event horizon of a black hole?

It's all about precision of measurement, after all. You can never know exactly where a given particle is, including the particles that make up your body. However you try to measure the location of one, there's always a slight chance that it's actually somewhere else. This chance drops off dramatically through distance, but it theoretically never drops off to absolutely zero no matter how far away you look. But does the event horizon of a black hole interrupt this, creating a hard boundary in where a particle's location may or may not be observed?

Honestly, I doubt this is answerable without a completed theory of quantum gravity, which has so far been beyond the reach of science. This is just one more instance where quantum theory and relativity theory disagree on what can/would happen, and until the two theories are reconciled, we can never really know what would happen in situations where they conflict like this.

But is any of that even necessary? You don't instantly die when you cross the event horizon. You can still carry on observing things as you fall deeper into the black hole. When push actually comes to shove for quantum immortality is when you come up against one of the black hole's several ways of killing you. (Which one occurs first will depend on the black hole's mass and maybe also which interpretation of the physics is correct. In some interpretations of physics, it might even depend on which frame of reference you're considering it from -- an outside observer might disagree with the unlucky test subject about which happened first.)

• Fall all the way until you're crushed into the singularity at the center?

• Spaghettification due to extreme tidal forces pulling your lower half harder than your upper half?

• Radiation damage due to time dilation effects blueshifting all the incoming energy from outside the black hole into deadly-powerful and concentrated x-ray radiation?

• Radiation/heat damage from crossing through a shell of energetic photons orbiting at about the level of the event horizon?

Quantum immortality would have to find a (statistically unlikely, but still technically possible) way for you to avoid all of these causes of death and keep you just barely alive enough to observe things. Radiation is the easiest -- just need to randomly divert enough of those radiation particles around you or tunnel them through you without touching, and you're safe from that. Not sure how quantum effects could protect you from the other two, though ... we might be getting back into the quantum gravity problem.