Well, it's still my computer, until such time as it comes to belong to someone else, but I think you mean to ask whether it's the same[/] computer. It depends on your definitions. Does an object exist in its own right, or is it composed solely of its component parts? Further, is an object's existence an inherent property, or derived from its functionality?
Let's consider a parallel case: suppose you're involved in an accident, and lose your left arm. Are you still a human being? Nearly everyone would say yes; you're a human being who happens to be short one piece. But suppose you lose all your extremities and some organs; you're reduced to the minimum possible fraction of a body necessary to sustain life. You're no longer capable of doing most of the things a human can; you're pretty much a vegetable. And yet, most people would still say yes; you may be a broken person, but you're still a person- the "vegetable" isn't mean literally. Now suppose we go a step further and replace all your missing parts with cybernetic equivalents. You've essentially become RoboCop (well, more likely RoboCostomerServiceAgent or RoboOfficeWorker, but stick with me). Your legs run faster, your arms lift more, your eyes see more sharply and the mathematical centers of your brain process faster and more accurately. Are you still human? Here, at last, we begin to encounter real dissidence. A thing is, at least in part, defined by its limits, and there's a notable difference between overcoming those limits and not having them to begin with (as those of you who become RoboProgrammer will undoubtedly tell you). But even those who don't regard you as (or regard you only partially as) human will still see you as a person. But this is because we look at human beings differently than we do machines (not that I'm condemning that!). So you need to answer the question: is a computer- is any object- what it does? Is it what it's made up of? Something else? If so, what?
It gets even murkier when we go beyond mere physical objects: the Catholic church doesn't have a single member or physical object that dates from the beginning of its 2,000 year history, is it still the same organization? It does have a few rituals, prayers, and ideas that date from Jesus' time, but nothing physical save a few locations. Further, some of its ideas have changed over time. Was it still the same church after Vatican 2? After any given Council that proclaimed an idea heretical or acceptable? Heck, was it the same church before Rome and Constantinople split away from each other (people will tell you the Schism took place in 1054, but that was just formalizing changes that'd been in progress for centuries, so you really can't say only one group was branching away from a continuous tradition)? When the Romans and Byzantines split from each other (people will tell you that the Byzantines were Roman, but if you buy that, I've got a bridge to sell you)?
Needless to say, these are broad questions that philosophers have been debating since time out of mind. But do we teach our teenagers of Kant and Wittgenstein? Do we bring up the ponderings on the great questions of existence, make their own lives less miserable as they realize that not everyone over thirty is an idiot, and at the same time make their own actions less insufferable, as they realize they're not trapped in a world entirely devoid of reason? No. We teach them polynomials.
