Don't confuse the end of Moore's Law with the end of computer technology progress. Moore's Law is actually a pretty specific law. It says the amount of transistors in the same amount of space doubles every 2 years. We're reaching the "end of Moore's Law", because we're getting to the point where we're close to making a transistor the size of an atom, which I think is pretty much the end of "doubling the number of transistors in the same amount of space every 2 years".

That doesn't mean that after we reach this limit we won't be making quantum computers, or use other materials like graphene transistors that can probably reach TerraHertz clock speeds, which may even improve at a rate of 2x every 2 years, too. But it wouldn't be "Moore's Law".

We're also working on "brain-like" computers, but I think these and quantum computers will become "mainstream", in a general-purpose way (not just for very specific tasks), in a few decades, which should be quite a while after Moore's Law dies (in the 2020's).

So the most likely thing is that we'll use other kind of materials for the same type of "classical computers" for the next few decades, even after the end of Moore's Law, but they will improve through increasing their clock speed or through other ways, rather than getting that extra performance from adding more transistors.