If you are someone who tends to get lost in Wikipedia (like me), time is a fascinating topic to branch from. Its very connected to other subjects, which makes sense considering how integral it is to our lives. Especially as science has advanced our ability to tell time, we have had to think more about what that means.
Fortunately, I am not overly concerned with metaphysics (right now); instead, I want to talk about a more immediate concern. Due to globalization effects brought about by communication bandwidth increase and latency decrease, people now exist who seriously defend the idea that we should adopt a singular, global time. Leaving out many details, the crux of the argument is, simply, that since converting timezones is difficult, we should do away with them entirely.
I personally know a handful of people who have done this: that is, set all of their clocks to UTC, kept their calendars in UTC, and lived their lives by UTC. (All of them happen to live in what we call EST, but that is an accident of where I myself have lived.) They have, to my knowledge, all stopped doing this, largely due to the vast inconvenience incurred.
Although one did it on a dare, the rest as far as I know tried it (for long enough to adapt) due to the theoretical benefits of a global time. If everyone keeps the same time, then there will be no more issues with temporal conversion, the argument goes. Furthermore, issues like timezones being artificially too wide would be mitigated since, rather than timezones being close-but-not correct, they will be wildly divergent. (My own EST is one such: the distance between eastern Maine and Indiana is about a third the width of the United States; another egregious instance is all of Central America sharing a timezone.)
None of which is true, it turns out. To the second point: while the conversion to UTC does demonstrate that it is possible to exist in an arbitrary timezone, their decision to revert reinforces the idea that it is strongly undesirable. More broadly, addressing the first point as well, what seems to happen is not that everyone can communicate, but that our words for times of day drift. Noon becomes 20:00, or thereabouts (or 8pm, if you insist). I do not know whether it would fragment to pre-train time (where every city had their own timezone), but it does not solve the fragmentation issue at all.
More problematically, however, we do need some fragmentation. Everyone in the world cannot expect to keep UTC 09:00-17:00 as a work pattern and see the sun (or at least, not enough). To at least some degree, whatever this fragmentation ends up being will be arbitrary; I can show that there must be at least two wake/sleep patterns for the general population, and that more would be good. As an upper bound, one per city we know to be too many. If there were some subdivision of a day that were handy to sync to, that would be nice; hours seem reasonable. But not all timezones align in integral values (have fun reading this Wikipedia article!). And all that said, I also recognize my own current inability to keep a 9-17 schedule (though I strongly appreciate not being required to do so).
The biggest problem my UTC-curious friends experienced was timezone conversion. Of course, since everyone around you is one timezone, converting to another, internal, timezone is work. But a few months in, it gets even worse: after learning the conversions to and from the local timezone, the local timezone moves. Daylight Savings Time begins (or ends, depending on when), and all the conversions change. Except that everything is actually worse than that: only some of the time conversions change, because not every zone has the same rules about savings time, not every country does it the same, and not even everywhere within a zone implements it the same way. (I encourage finding more about this at your local Wikipedia!)
And while it is of course a larger problem for those in UTC (as most things are), Daylight Savings Time is a problem for everyone. The problem could be mitigated somewhat if every zone were to implement it at the same time, but this is nonsense: one hemisphere will have crazily incorrect time, and while this is technically better, there is little incentive for it. Much more appealing, I think, is the idea of abolishing Daylight Savings Time entirely. I will leave it to others to better make the case for this.