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Here are some escape time pics of a power-2 Mandelbulb with the Mobius cylinder rules applied. Same configuration as in OP. I also tried inverting the 'supplementary' component (I don't know if the Mandelbulb's 3rd dimension has a name, I just call it supplementary) in addition to inverting the imaginary component, but it didn't change much.

Top view


Side view

Nice!  Wrapping in the other direction looks good too, just change the formula line to (I like this version with the width slider around 1.3).  Also, using the regular quadratic Mandelbrot set distance estimator seems to work well enough.  I tried some experiments by rotating instead of reflecting, basing the angle on the continuous X value, got a sort of sinusoidal strip with cusps opposite needles alternating up and down but it wasn't as interesting.

Not very mathematically satisfying, as Levi mentioned, but the pictures look cool.  A Möbius strip isn't orientable, and squaring complex numbers has to do with doubling orientation relative to a distinguished origin (which also isn't really possible to pick on a loop).  But, it might be worth trying some iterated function systems (using the chaos game method) on a Moebius cylinder: http://math.stackexchange.com/questions/1369588/is-this-a-valid-example-of-a-non-euclidean-sierpinski-attractor