It is one hour to midnight here in Finland, right now, when I write this. If I was in Morris, Minnesota (to pick a fringe famous place at random), it would be three hours after midday instead. That's because what a clock shows is not time, but a sun tracking system. It's a noon when the sun is overhead, or as high overhead as it's likely to get any given day.
Go out into space, and you'll see the time zones are just an illusion, though a useful illusion. It makes no sense to note the Sun being overhead when there's no turning ground beneath your feet, to make that overheadness come around at regular intervals.
If you landed on Mars, you could call the moment the Sun is overhead a midday, 12 am, too; but the next one wouldn't be along when your clock had turned 24 full hours. (You'd need to wait for an additional half an hour or so; a Martian day is around 24.5 hours.)
Worse still, if you carefully measured time from the 15th of September, 2011, to 15th of September, 2012, you would not be back in the same Martian season, no matter whether you counted days by units of 24 hours, or by suns going overhead. Years are an illusion just as much as days. Mars days and Mars years, if you define them by the planet, are not the same as those of Earth, not by length or by how many each contains of the previous unit; but if you don't define time by the planet you're on, you're in for a lot of practical trouble:
"Let's meet here at noon tomorrow."
"What? Noon tomorrow is in the middle of the night!"
"But the bus for the spring festival leaves at noon!"
"And the spring festival's in bloody December! I hate Mars!"
But if you use the local units... well, that means every planet has its own days and years. (Don't think about the moons. You'll break your mind.) Their own months too, probably, because the length of the year influences how you divide it into manageable chunks. (Our months aren't moon-ths anymore. Full moon to full moon isn't a nice integer number of days.) The Martian year is a little less than 669 local days (measured by the Sun); since the closest multiple of 30 is 660, you could start with 22 months of 30 days, and add a day to a few of them --- or you could start with 12 months of 56 days and whittle a few of them down. (Do you want to have 12 months for the old names' sake? That's a swindle waiting to happen --- "Just work here until December's over, Earth boy.")
Worse still, what do you do off-planet? Say you have a colony that floats somewhere in the asteroid belt. There you have no natural unit of a day; even if your colony rotates, that doesn't need to be the source of your overhead illumination and rhythm of life. Your colony goes around the sun, probably, but that doesn't create seasons in space. (Well, your distance from the Sun may vary, but that's unlikely to have any immediate effect.) Such a colony would probably use the "home planet's" days, months and years, and schedule their lives accordingly, at first.
But what about when the space miners complain that 24 hours is too short to contain both sleep, travel and work? (Or complain that all that one can psychologically bear to do is 4 hours of work without rest, but 20 hours of free time is economically impossible, so could they have an 18-hour schedule?) Why, that would be a temptation to retool the already arbitrary calendar. (Even more so if the home planet grew distant, and not particularly politically dominant. "Down with Earthling tyranny and the oppressive/slothful Earth day!")
Ah, but that runs into something that I know nothing about: the human animal and its ability to adapt into new day lengths. Given suitable incentives of light and darkness, could you adapt to 30-hour days? Or 12-hour days? There's probably some upper limit where one just needs to rest and sleep (whatever sleep actually is, and whether it's just another condition awaiting a cure; I don't know), and there's probably some practical lower limit where shuffling day and night, sleep and waking, becomes just silly; but what are the limits of the beast? (Related: if you have a 30-hour day, how many hours of it do you need to sleep to feel rested? The same fraction as of a 24-hour day, or more, or less?)
And what are the effects on visitors?
"The pilot wishes to welcome you to Asteroid Xena-Callisto. The local time is 12:25, the local day is 16 hours, of which the first ten are allocated as night. The food court is open; the customary dinner crush is at 15:30, and the breakfast at 10:30. We hope you enjoy your stay. Those not arriving from modulo 16 timeframes plus minus 4 are encouraged to contact the Asteroid Time Consult for advice, sleep pills and a temporal lightbulb/calculator."
What are you left with, then, with different days and years? Well, hours, minutes, seconds obviously... but an hour is of such length that 24 of them make a full Earth day; minutes such that 60 make an hour, and seconds so that 60 make a minute. What if your planetary day is 22 hours, 31 minutes and 12 seconds? Woe the clockmaker! Woe anyone, actually, that tries to measure anything by such hours --- 48 hours will not be any integer number of days; no decent integer number of hours will be. The clocks will have 22 hours, and a half, and a bit more; that's all kinds of inconvenience.
So there is a great temptation to redefine hours too: to contain a different number of minutes, maybe. And if that is not enough --- "Our day is 1500.5 minutes!" --- then even the minutes could be redefined. The Earth day is (he said with some handwaving) 86400 seconds long; unless you are damned unlucky, your random planetary day could be measured in seconds, and divided into manageable chunks. ("What? Our day is a prime number of seconds long? Sweet bleeding cloaca of Constantine!")
Infobox: Even the Earth day is a little bit wonky. Have you heard about leap seconds? Since 1972, 24 seconds have been added to our atomic clocks to keep us up with time as measured by the position of the Sun. Leap seconds are in use because they're at the right level of coarseness: they're needed one or twice a year, tops, and nobody really notices when they're inserted into the count of time. Time that needs to be nudged like this isn't bothersome; indeed, most people don't even notice it. (Do you think your neighbor has heard about leap seconds? Hearing, would he or she say, "Seriously? There's leap seconds in my clocks? I've never noticed that thing!"?)
Our current way of counting time could no doubt be improved: 60 seconds to a minute, 60 minutes to an hour, 24 hours to a day is (to me anyway) a messy system. How may minutes in a day? 1440? Yeah, that's nice and obvious. How many hours in eight days? Hmm? If you ask me, units of time should be easy to convert, possibly just by moving the decimal separator, instead of being crawling horrors of Sumerian astrological geometry. (What? You thought the sixty-sixty and so on divisions of time came from some cool, rational consideration? Heh!)
Also, the whole base-ten system of numbers? That we talk of one hundred (ten-tens), two tens and three ones, and write 123? A blind monkey with a math degree could do better; who designs a numerical system based on a number with just two factors? A blind monkey without a math degree? Compare just 10 and 12. Ten is 2 times 5; ten marbles can be divided into either 2 or 5 equal piles. Twelve is 2 times 2 times 3; you can divide it into 2, 3, 4 or 6 piles without a knife. Having a number system with good divisibility is kinda important in practical life. ("Honey, I have 1oo muffinses and 6 guests. How do I divide them?" vs. "Honey, I have 84 muffins base twelve, and 6 guests... oh wait, 8 twelves is 8 twice-sixes, so 16 muffins each, and then 4 are left over. Base twelve is so simple even a househusband can do it!")
An hour of 100 minutes would be ugly: half of it would be 50 minutes; a third something unseemly; a fourth 25 minutes; a fifth 20 minutes, a sixth, a seventh and an eighth unspeakables. Fractions are important; and the decimal system is not particularly good with fractions.
Compare an hour of 60 minutes: a half is 30, a third 20, a fourth 15, a fifth 12, a sixth 10 --- counting and dividing something by sixties and their fractions is so fine even ancient Sumerians saw it! I blame the Egyptians for our decadic foolishness; the Egyptians, and biology. (If only we had six fingers on each hand, then base twelve would come naturally. But no --- the wrong gnathostome got crushed in the Ordovician, and five fingers it is.)
Now, this talk of different bases and different counts of time is blowing hot air, right now; there's a wee bit lot of practical stuff that is against an alteration as fundamental as these. But suppose that we one day get into space, and there's a bottleneck of a first colony, a few thousand people adjusting to an already foreign environment, and someone adds one adjustment more --- why, our distant children could laugh at us and our weird ways of handling time and numbers. Or, on widely separated islands of life, there'd be space for speciation of numbering-cultures --- have whatever expression that most pleases you.
And in the future, the handling of time... well, your mobile phone probably keeps time by querying, now and then, a server somewhere. That server might take time very seriously; and your mobile phone might actually keep you up to time with the accuracy of leap seconds; but it's not likely you notice that. Could be that in the future changing a time system is no more trouble that changing a time zone; just one of the small problems of traveling, something mainly fixed by tapping your watch-equivalent digital computer thingie, and by griping: "Oh man, this rocket lag is killing me. How can it be dark again?"