Quantum immortality was a concept in quantum mysticism invented by Hugh Everett, the guy who originated the Many Worlds Interpretation. It's not even taken seriously by defenders of Many Worlds. Major proponents of Many Worlds like Sean Carroll even admit it is nonsensical and silly.
Imagine if a company perfectly cloned you. If you then died, do you expect that your consciousness would suddenly hop into the clone and take control over them? No, it makes no sense. The clone is effectively another person. If you die, you would just die. The clone would keep living on because the clone ultimately isn't you.
The obvious problem with quantum immortality with it is that if you truly believe in Many Worlds, then the other branches of yourself in other copies of the universe are effectively like clones of yourself. You dying in this branch of the multiverse doesn't somehow magically imply your consciousness can hop into another branch where you are still alive. "You" as in the "you" on this branch where you die would just die, and the other "yous" would continue to live on.
Penrose's ideas are not taken seriously either, because the arguments for them are comedically bad. Pretty much all physicists are in unanimous agreement that quantum computing needs to be well-isolated from the environment and incredibly cold, the opposite of a human brain, and so there is zero chance the brain is utilizing quantum computing effects.
Penrose's argument is, and I kid you not, that it is possible for humans to believe things they cannot prove, for example, we cannot currently prove Goldbach's Conjecture but you can choose to believe it, and therefore he concludes human consciousness must transcend what is computable. Since no algorithm can compute the outcome of the collapse of the wavefunction with absolute certainty (as it is random), he then thinks that the human brain must therefore be using quantum processes.
I genuinely don't know how anyone can find that argument convincing. The barrier towards creating artificial intelligence obviously isn't that AI has a tendency to only believe things that are rigorously computable. In fact, it is quite the opposite, AI constantly hallucinates and makes statements that are obviously false and nonsensical. The physical implementation of the neural network can be captured by a rigorous mathematical model without the output of what the neural network does or says being all rigorous mathematical statements. There is no contradiction between believing the human brain is not a quantum computer and that humans are capable of believing or saying things that they did not rigorously compute.
Penrose then partnered with Hameroff to desperately search for any evidence that there are any coherent quantum states in the brain at all. They start with their conclusion they want and desperately seek something out that might fit it. All they have found is that there might be brief coherent quantum states in microtubules, but microtubules are not a feature of the brain, but of eukaryotic cells generally, and they play a structural role as a kind of lattice that keeps the cells together. Even if they are right that microtubules briefly can have a coherent quantum state, that does not get you one iota closer into proving that the human brain is a quantum computer in the sense that coherent quantum states actually play a role in decision making or conscious thought.