Square Face with Round Time is an analogue clock with a
completely square face.
Here's how it works: Taking advantage of an idea I
posted a while back (link) Square Face clock features
telescopic hands.

This time, the ability for the hands to elongate and
contract in length has been
motorised and linked to their
circular motion. This means that they constantly adjust
in length as they rotate, resulting in them always pointing
precisely to the numbers enscribed along the perimeter
of the square. By doing so they simultaneously describe a
circle and a square.

De-luxe variations feature oblongs and other odd shapes
yet with the hands always pointing out to the edges of
the face perimeter.

One contradiction is interesting, but two combined movements may cancel out the effect. It's all interesting, ie the idea of drawing a square on a continually rotating piece of paper.

I feel like extending the hands using individual motors is
inelegant. If you used a guide track leading a pin attached
to the telescoping segment of each arm, I think you could
get a fairly good approximation of a square, driven by the
clock mechanism.
You'd probably need to round the corners at least a bit, but
the overall error would be good, and the motion would be
superior.

//Or you could cheat. A standard clock mechanism behind a
panel with a square glass cutout.//

I was going to say that wouldn't work for the (shorter) hour
hand. But you could do it /twice/.
And you don't really need the glass, just the square hole.

If you made a series of gears that, at their circumference,
contained other gears, which at their circumference,
contained other gears...for the appropriate number of
iterations (6 seems to be a minimum for a passable square), you should be able to construct a Fourier object
that traced out a square (or any other shape, see links).
From there, you could attach some black elastic back to the
centre which would serve as the "hand". The tension would
likely make this impractical, but it's worth mentioning just
as an excuse to post a link to the Fourier stuff as a
tangential thingy.

There are many mechanisms and ways to make the hands
extend & retract; the trick will be squeezing it all into the
(generally minimalist) hands & centre area of the clock.

You could have the hour hand attached to a chain (or toothed
belt) that runs around the perimeter of the square clock with
small sprockets at each of the four corners. In this way the
hour hand would almost always be vertical or horizontal
(except when moving around the corners). Another chain and
sprocket set behind (or in front of) the first set could have the
minute hand.