Well, I just can't stop thinking about this time stuff. Here's a bit more that I conjured up:
When the first settlers land on Mars for the purpose of permanent residence, they can establish that date as the first Martian day of the first Martian year. Each month would have approximately 55-to-56 Martian days. (If Mars' orbit around the sun is more elliptical than Earth's, then some months would need to have significantly more days than others.)
The designated length of a Martian second would be about 2.8% longer than an Earth second, so that each Martian day could be exactly 24 hours, 0 seconds, under this local timekeeping system. It would be one day-length different from Earth about every 36 days.
If they land within a short interval of time before the northern solstice on Mars occurs, then their arrival date would be called June 1st, 1. The northern solstice would happen sometime in June, the southbound equinox in September, the southern solstice sometime in December, and the northbound equinox would happen in March. The final day of the year (i.e., Martian New Year's Eve) would be something like May 56th. The next day would be June 1st, 2.
Now, instead, if the pioneers first arrive on Mars about halfway between the time of the southbound equinox and the southern solstice, then they could call that date November 1st, 1, which would be the first day of the year, instead. The solstices would still occur in December and June, the equinoxes would still be in March and September, but in this case, the Martian New Year's Eve would take place on October 56th, or so. The next day would be November 1st, 2.
Whatever longitude the first permanent settlers land at would be deemed the prime meridian, and the 24 Martian time zones would flow from there.
So that's what I thought of about how "local time" on Mars will work. Fun to ponder.
However, this kind of local calendar system won't be very practical on Pluto, since each month would need to have about 7,000 days. (ha)
Same for any exoplanet that orbits its dwarf star every 15 days...as each calendar month would just have 1 or 2 days.
So an alternate calendar scheme would need to be devised for cases like these.
Also, 24 hours for a solar day won't be very practical if the planet (or moon) rotates very slowly, such as Venus, Titan, or the earth's moon. And especially not so if it's tidally locked to its star, which Proxima Centauri B might be.
So the experts would have to figure out an hourly time frame that could work for these worlds, as well.
Just my thoughts.