[KboPAacDA3]

The key reason is the Coordinated part. By letting the Moon handle it's own timekeeping, it reduces the hassle of getting regular updates from Earth clocks. Earth-based atomic clocks will continue to measure its own precise second lengths and take measurements of quasars to get accurate wall-clock time. Moon-based atomic clocks can have their own separate network of atomic clocks and measure the same quasars to get the same wall-clock time.

[ivan_gammel]

Maybe not just that. We generally need to establish the time of events on another celestial body and process that time locally. Having separate clocks is trivial, but recording correctly sequence of events that happened a few light-seconds away is interesting, when taking all effects into account. Should we record the local time of event? The observer time (=local time+distance+gravitational effects)?

[jenny91]

Do we have any atomic clocks on the moon at the moment?

[mikeyouse]

Looks like "no" but a few years ago they launched / tested the "deep space atomic clock" which seems like it'll be the basis for future space-based clocks.

https://en.wikipedia.org/wiki/Deep_Space_Atomic_Clock

[zokier]

LN-1 that landed recently might have an atomic clock onboard. But the details are scarce so I'm not completely sure.

[kspacewalk2]

Step 1 of N for Lunar Independence.

Free Luna!

[euroderf]

Certainly lunar clocks should feature on the first series of lunar postage stamps.