I'll assume that you know what the dewey decimal system is.
A clock that runs in dewey decimal is less obvious, but really quite simple. The time is, in its standard form, numbers. The dewey decimal system also uses numbers. So all the clock does is convert the time to a three digit number (9:03 matches up with the dewey decimal category 903).
Naturally, some accommodation must be made for any time later than 9:59. For these, I simply add the middle two digits (10:29 produces 129, and 12:52 produces 172).
It may not ever become a widely used system of time, but it has some of the character that the usual digital clock lacks.