Curiosity's computer tracks the number of seconds from an epoch [launch[January 1, presumably - Hobbes]2000 CE], landing happened at 397502503. The mission calendar uses sols (Martian days) counting from the landing day = sol 0.
(from The Design and Engineering of Curiosity, Emily Lakdawalla)
I see no reason not to use Earth time as the baseline in a Mars community. Anything else would massively complicate things for no benefit.