Unless there is a spiritual 'you' that maintains a connection to your physical body and does not attain a connection to the computerised copy.
OK, give me even a remote reason to assume such a thing exists, and then we'll talk.
Given the lack of decisive proof on the topic it seems to be a valid argument...
You're not familiar with the flying spaghetti monster, are you?
And of course, the 'you' isn't completely random, it is a progressive process that is the result of a series of comparatively mild events, a sufficiently violent alteration could be perceived as severing the progress and creating a new series.
Copying your consciousness onto a computer (one that could also simulate all the hormonal and perceptual processes of your body) would change your consciousness less than, say, having a stroke that wipes out half your personality, or a bout of amnesia. But while nobody would argue the stroke or the amnesia change "you," lots of people think copying the consciousness would.
It's essentialism. It's a part of the way people inherently see the universe, and is fine for that purpose, but when investigating things scientifically, it has to be discarded.
Of course the real killer is the fact that the copy and the original could co-exist, it isn't easy to persuade someone that the person they are looking at is themselves...
The person isn't "themself," but not any less so than "themself" from a year ago is not themself.