Consider a thought experiment formulated by Leibniz: suppose you have the option to choose to have all the riches, talents, fame, good looks and lovers you desire, but on one condition: that you forget absolutely everything about yourself up to that point. Would you do it? If you answer is no, that implies that you think that whatever you are, your conscious experience and memories seem to be a necessary part of being you. So, no memories = no you. The new rich, talented, beautiful person would effectively be someone else. Imagine then a case of assault in which the victim loses all memory. Should this now count as murder?
In the following presentation, Julian Baggini explores the question of the self, whether it exists, whether it's an illusion, and whether we should understand an illusion as something that's not there, or as something that's simply not quite what we normally take it to be, but which is there nevertheless.
And if you want to listen to the whole thing, including an interesting Q&A, you can listen to it here:
For more on this issue, check out the Brainspotting series.