Andrew Gideon wrote:
I understand that mode S transponders have unique identifiers,

True, they not only have unique identifiers but can also transmit
a settable identification number (such as a flight number).

thus
eliminating the need for beacon "codes" as we use them today.

It might, but ATC still operates in a discrete code based world.
I've got mode S but it hasn't changed squat with respect to ACTC


So, in an ADS-B world, what's the story on transponders. Is mode C still
required? Does the ADS-B UAT replace a transponder? Something else?

Transponders are still required because ATC is ignorant about ADS-B
and so is TCAS.