ZFC is only "standard" because it (or conservative extensions of it like Universes) allows us to do pretty much whatever we want. However, recent developments in higher category theory have led some to call for a new "set theory" (mentioned above) that categorifies the classical theory of sets and cleanses the "evil" from it. (Evil, of course, in the sense of considering objects up to equality rather than isomorphism.)

It seems unlikely, to me at least, that someone will be able to develop a useful alternative set theory that isn't equivalent to or stronger than ZFC, the operative word there being useful. That is, we don't want whole edifices of mathematical thought falling into the ocean under our new set theory.