It was recommended multiple times in history, the problem is that it doesn't really solve any problems, just moves the problems elsewhere:

• At some parts of the world "midnight" (the time when the day changes) will be during the "day". Would you like that you have to use a different calendar date at the morning and at the evening? It also makes much harder to check if something happened one or 2 days ago by simply checking the date, you would have to know the new UTC time as well
• You still would have to know how far they are from you to set up an international call. Some people wake up at 2:00 UTC, some wake up at 16:00 UTC. So actually nothing solved, but you have to use different numbers instead of time zone names.
• Time in China works like that, you can already see how it's going there. Full of China is one time zone, on its western border if you would cross to Afghanistan you would have to set your clock 4 and half hours backward. In Xinjiang solar noon is around 15:00 (3 pm). How people live like this? They simply use different timetables, 9-5 job is something like 11-7. So even it doesn't have a separate time zone people live like they would have a separate time zone.
• Swatch Internet Time was a well known example of this in the late 90s, you can read the general problems with it on wiki

But then we would need some method to quickly discern the relative position in the day/night cycle of a locality, perhaps some form of number to indicate the percentage of a day before or after midnight their local area is at. Then probably just add that offset so that the viewer doesn't need to do the math everytime.