There are a few mentions of how time is kept on the moon in "Artemis" by Andy Weir. He does his research pretty well and is a pretty comprehensive researcher, so his ideas are probably worth consideration.

One way time is kept is through the use of "Earth Time," which is the time on Earth. Most residents of Artemis, the city on the moon where the story takes place, keep track of Earth Time so they can stay in sync with the rest of the world.

There is also a "Lunar Standard Time," which is a time zone that is based on the location of Artemis on the moon. This time zone is used by people who work in the moon's tourism industry and by some of the government officials on the moon.

Finally, there are also references to the use of "mission elapsed time" (MET), which is a system used by astronauts to keep track of the time since the start of a mission. This system is used by some characters in the book who work for the lunar mining company.