What's more likely, in practice is an expansion of self into hardware.
Basically, if you bolted a computer to your brain, you would still be you, just with better memory etc. If, at this point, your brain died, then you die. However, if you kept adding to the computer side, more and more of "you" would be software based. What happens at 90% (10x capacity) or 99%? If your brain were to die, "you" only lose 10% of your capabilities. So long as everything critical is duplicated in software, the pure software version of "you" can go on thinking.
Critically, there is no neuron duplication here. It relies on both the plasticity of our brains to adapt, and the fact our self is continually updated.
Interestingly, the first cases might not even be truly planned. What happens when your "neural AI assistant" can continue to function after your death. If it is self aware, and believes it is you, at what point is it still you?