What would be easier with our conversation? I guess we are at the other side of the globe. I wasn't answering because I was sleeping, you won't answer immediately because you are sleeping, what would help if our clocked showed the same? Both of us sleep because the sun is down, not because the clock says something.
With long range space travel we will have to deal with the relativity of time as well. Lets say we define an Epoch and both of us do something exactly a googol second later but we live on a different planet, it wouldn't be at the same time because of relativity. If I would say lets do something when it is 12:00 in London, UK, Earth, than you can convert it to your relativity and time zone.
About the intetplanetary usage of the current system, it's already in effect, LTC aka Coordinated Lunar Time is already under development since this April.
On the moon, a smaller body where the gravitational pull is much weaker, time moves more quickly and unevenly: Lunar time gains about 58.7 microseconds per day compared to Earth’s time, though even this can vary, depending on the altitude and longitude where lunar clocks may be located.
“An atomic clock on the moon will tick at a different rate than a clock on Earth,” Kevin Coggins, manager of NASA’s Space Communications and Navigation Program, tells the Guardian’s Diana Ramirez-Simon. “It makes sense that when you go to another body, like the moon or Mars, that each one gets its own heartbeat.”
So the solution is to just convert between time zones, and resync them sometime because they can drift because of relativity.
I don't know how a base 10 day would work, it just annoys me that its reform wasn't successful with the metric system. But I don't know how it will work when people will live on Mars. A day on Mars aka a Sol is 24 hours and 39 mins. Will they have "Mars hour" which equals 1 Earth hour and 1.6 Earth minutes? If not, they could simply use Earth hours, than it won't be possible to create classic circular clocks.
But I have to google when the sun comes up for you, it's literally the same thing.