This is a personal project, I wanted to do a watch which would turn square.
I got my calculator and my criterium out and decided to solve this existential problem.

Here is the Eclipse watch, wich the hour clock turn square. 
It’s a solution using a standard mechanism.
This is a watch with a simple archetypical form which discreetly highlights the square rotation mechanism at it's center. Touches of red are used on the clock hands to read time with accuracy and precision. Markings are engraved on the inferior part of the glass.
It is the minute clock hand which drives the system with a small gear. The piston gear is three times bigger than the minute gear which allows the piston to reach the appropriate speed : four laps every 12 hours, the clock actually turns square.
Here is a little video to better understand the kinematics of the watch:
As for the appearance I was inspired by navigation measurement tools, especially the compass and the symmetry of its needles.