I am a proponent of a Martian timekeeping system where each day has 24 hours 39 minutes and 35.244 seconds and all the rest of time units have standard lengths. For example, at midnight your martian clock would show the time as 24:39:35 for a fraction of a second, then wound back to 00:00:00. The calendar then would track the Earth calendar as close as possible, with shortened months to accommodate for the longer day…
Wouldn't that get incredibly confusing with the calendar always being slightly off, despite using the same names? At that point why not just stick to a integer day count for the date. Having a non-integer number of hours in a day might also get confusing. It would be near-impossible to make an analog clock, for example, and stuff like dividing up a day into three shifts would be really cumbersome. For day-to-day life…
I think you can keep the calendar within +/- one day of a certain Earth location by carefully adjusting the length of the months. So it would be much like the dates vary on Earth between time zones, for example at the time I'm writing this the date in Hawaii is March 1st, while my local date is March 2nd. But less predictable, yes.