The dialogue we are reading is ostensibly both about immortality and personal identity, but the main philosophical issue is indeed personal identity. The two are related, of course - one cannot easily discuss immortality without facing daunting personal-identity related challenges; but that is a different matter, because here the direction of the discussion starts from personal identity. And this is well worth the effort, considering that personal identity is one of the most interesting and intractable philosophical problems.
First of all, you should be clear about something basic - moreover, something that most philosophy students tend to confuse. As a philosophical topic, personal indentity is NOT about pscychology, or about character, or about mid-life identity crises. Personal identity is a logical problem: What is that elusive X [you, for instance] that persists over time, and which, by virtue of its persistence, makes it meaningful to talk about one and the same entity [you, for instance] as being there and even undergoing changes? Sure, you change over time; but we still can make sense of saying that it is you who changes. So, what is persisting over time - what is personal identity?
You should not confuse personal identity [A is identical with itself], with indiscernibility. Two entities, A and B, may be indiscernible in all respects. This is actually controversial - after all, A and B cannot be indiscernible in all respects since they are not the same. At any rate, if A and B are indiscernible, we still refer to them as two different entities, A and B, which are exactly similar but not identical. A and A - A and itself - on the other hand are one and only one thing. In other words, from a logical point of view, personal identity presupposes necessarily 'absence of competition': there can be no other A, there can be no other you. So, it is a logical matter, as well as an empirically defensible proposition, that twins and clones are TWO different entities even if they are, arguably, exactly similar in all respects.
There is a paradox associated with personal identity. Every person changes over time. One acquires new characteristics, and loses old ones, all the time. How, then, can a person remain identical with itself over time? Shouldn't one be identical in all respects - indiscernible - throughout or over time?
Personal identity problems have been known to thinkers for thousands of years. Here is an ancient riddle, the 'ship of Theseus.' Theseus' famed ship keeps rotting away after having been sea-borne for so long. Theseus keeps replacing plank after plank. One day, he realizes that not a single plank from the original ship has remained. Is this still the ship of Theseus? At what point did it cease to be the same ship, if this is indeed what happened? Suppose that an enterprising entertainer had collected all the discarded planks and has now put together - rickety though it might be - what he presents as the 'true ship of Theseus'. Which one, if any, is the real ship of Theseus?
In the case of human beings, PI [personal identity] seems to relate to certain reasonable criteria: continuous memories, appropriately caused [not induced by hypnosis, for instance], which attest to a continuous and uninterrupted sense of one being the same person over time; a certain gut-level awareness of being the same person throughout, waking up every morning as the same person one was before; a test of anticipation of pleasant and unpleasant experiences to be expected; certain personality traits. Biological markers cannot be that important - after all, DNA can be cloned but, as we said, earlier, the clone is numerically distinct from the original person, and, hence, personal identity is preserved only for the original person - the clone is a separate person. Appearance can and does change over time, not to mention the increasingly sophisticated and thorough techniques of plastic surgery that are available. Cells renew themselves, at different rates of course but ineluctably over time. But, are the less tangible markers reliable? What about amnesia? What about hypnotically induced suggestions?
With the above in mind, you are almost ready to investigate the problems presented in the dialogue you read. Should GRetchen, the philosophy professor in the dialogue, accept to undergo a brain transplant? Also to consider are hypothetical cases involving time travel and brain replicas? A brain replica - a phantastic scenario, of course, this time - would also have the same memories and awareness as the original. It seems that only the strictly logical criterion applies at this point - the criterion of numerical identity, i.e. that there can be only thing that is identical with itself. Is this sufficient, though?