This is in the same area of exploration, except that my main requirement is that, visually, the QR code is a random-dot stereogram. If you align your eyes the right way while staring at it, a 3D image appears.

Other requirements, like that it decodes to something that is meaningful, are secondary.

Alok's quine work is interesting. The actual quining is trivial because the program pulls out its own image using Location.href. (The trick seems to be that when JS code is embedded in a literal "data:" URL, then the Location.href can pull out that URL itself. I'm not a JS person, though.) It looks like the hard work in this project was developing the QR code encoder using a verbose OOP approach, and then code golfing it through 25 steps to make it small.