I have a vague image from the 60s of a rather obscure non-fad (as in it never really caught on). I must have seen it once or twice in a movie or TV show. The concept was a thing that looked like a painting – abstract or minimalist – where you could tell the time based on the components in the picture, like counting the number of birds in a line to figure out the hour.
Does anyone here remember anything like that?
I vaguely remember a Odd Couple (I think?) episode where the apartment had been redone with ‘hip with-it’ furniture. (I think there was a chair that was in the shape of a hand.) There was a clock that was some assortment of lights that no one could figure out how to tell time on. The of course, the dumb character glanced at it and said “Oh, it’s a quarter to three, I gotta leave!” Or something like that.
But I don’t think I ever saw one out in the real world.
There are some cute Salvador Dali-style “soft clocks” – here’s one on Amazon – which are just as cute as anything.
I did some Googling, and, alas, all I was able to find were standard clocks, with really surrealistic or abstract art backing. The usual long hand and short hand and second hand – in front of Pollock or Mondrian or whatever. Not what you described.
I come across timepieces that show the time is odd ways every now and then. Here’s an example:
http://www.gadgetsandgear.com/binary-clocks.html
I was surprised not to find more on the web. But frankly, not disappointed. I’ve also encountered unreadable watches.
I’m a big fan of art clocks and have been since I was a kid and saw that Odd Couple episode mentioned above. ETA: aye, they had a chair shaped like a hand.
Oblique Clock - this one is an easy read once it’s explained to you, but it still looks cool as hell.
I’m a big fan of this Mondrian-style clock but I haven’t got one yet.
Oh, and here are some older model art clocks from the same manufacturer.
I think this one would be fun as hell.
I have a pi clock, she said smugly. It goes counterclockwise (!) and tells time in radians. 
But that’s not much help in answering the OP’s question, is it? Are you thinking of things like this Op Art kinetic wall clock?
I remember several designs from Pop Mechanics, by artist Roy Doty, that used a special offset drive motor… one spindle for hours, another twelve inches or so over for minutes. The dials were gears, or flowers, or ladybugs and you had to know which ones to look at among many. It was suggested as a party trick that you tell time on demand without a watch or visible clock.
Oooh yeah, Roy Doty picture clocks!
When I think of '60s clocks, I think of sun clocks.
I prefer the notion of a picture clock that you can glance at and get a sense of about what time it is rather than something that you have to decode. Like, ah, I see the lion, so it must be after four, but the trees have not sprouted yet, so it has to be before six. Or something like that. The random Mondrian is kind of good that way.
Of course, being a fan of maps, I think it would be cool to have the Geochron clock/calendar thing, but it would be even better if they could do it in a globe.
This was the first thing I thought of too! Here’s the scene on YouTube. Actually its the whole episode if you play it from the beginning. And the dumb character was Penny Marshall’s Myrna…
That’s them. Thanks for the ride in the wayback machine… those pages are engraved on my mind.
