It's hard to answer, because I'm not neck deep in the design of computer chips.

From a physics standpoint, silicon is great because it's a semi-conductor. Not only that, but doping silicon is "easy" and silicon's crystal structure allows for more doping than other semi-conductors (but not all).
Pure silicon conducts at 0.7 V, which is a decent sweet spot between not so sensitive that random noise or thermal motions will trigger it, but still "pretty low."
Silicon oxides are also "easy" to make and are used as electrical insulators which are stable against degradation by water / steam and resilient at temperatures in the many hundreds of degrees.
The Earth's crust is basically made of it, so it's readily available.
It's chemically stable and non-toxic.

It's kind of a miracle element for making electronic devices.
It's also hard to say how much more we can do with it. Seems like every few years, there's a new use for silicon in the electronics manufacturing industries, which only increases its utility.


So that's a lot a new material has to compete with.
I'm pretty confident that whatever replaces silicon will not be a pure element. It'll be some breakthrough from materials scientists finding some lattice of different atoms that allows for a significant improvement in some desirable property.