Here's a fun thought experiment – imagine slowly, individually replacing each neuron in your brain with an identical copy, one at a time. You wouldn't notice each of these (a similar process probably happens throughout your life). Is it still "your" consciousness at the end of this process?
Now imagine that instead of an identical biological copy, we use some kind of technological copy that perfectly mimics the biological in I/O (of course, you must imagine we've invented such a thing). Now at the end of the process, your brain is completely electronic. Still "your" consciousness?
Now, take it one step further and imagine that instead of a technological copy that mimics I/O, we instead (slowly, one-by-one) hook the terminals of each neuron to some kind of transducer that allows us to simulate that nerve in a computer, but deliver the I/O back to any neighboring biological neuron (again, you must imagine we are able to do this). By the end, your brain is completely simulated. Still "your" consciousness?
It's difficult to say at which point in this progression the "you-ness" is lost. But I think most people would agree that it _is_ lost, at some point.
(FWIW, I realize that this argument is somewhat isomorphic to a Ship of Theseus – with a little bit of techno-dysphoria mixed in).