I really love this question. Were we to establish a colony on our moon, we would probably want a calendar/time system which actually corresponds to physical patterns and cycles on the moon, but also something which is easy to track and keep in sync with Earth, since the two civilizations would certainly be in close contact.
In a way, we already do something this here on Earth with time zones, though time zones are just offsets within the same second/minute/hour/day system, so people throughout the world are in different times, but they stay in sync. But beyond that, our entire calendar system (we currently use the Gregorian Calendar), is mostly based on our own planet's positions relative to the sun. It just wouldn't make sense on the moon. Earth days on the moon would just be arbitrary 24-hour blocks of time; they wouldn't correlate at all to whether it's light or dark outside.
So, if we really wanted to keep some logical synchronization between earth and moon calendars, we would probably want to adjust here on earth to a lunar-based calendar. Most older civilizations did this, using a full lunar cycle to indicate a "month." But here's the (cool) thing: the moon is a little lopsided and way gravity works, the exact same side is always facing us. We never see the back of the moon. So, it's rotating at exactly the same speed as it's revolving around us, meaning a "day" on the moon lasts one full lunar cycle: just over 29 earth days. So even the idea of a day, how long a day is, and what you could accomplish in a day is so far from Earth's that it's hard to even compare the two.
Maybe one idea is to have weeks or months here on Earth which would line up with days on the moon. Earth would still have its 24-hour days and the Moon would have its 709-hour days, but at least we could predict Moon time using our own calendars. One proposal is the The Hermetic Lunar Week Calendar which continually adjusts the length of a week to keep the months in sync with the moon, so that a full month is always one lunar cycle (or lunar day). It would be weird but in a future where humans are legitimately split between Earth and Moon, it would probably make sense for both to keep a calendar like that, which takes into account the length of a day on both bodies.