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iIf you draw a little square in the C-plane completely outside the Mandelbrot-set, the corresponding iterations in the Z-plane will be a little square jumping around, getting bigger and more distorted.  But if the square contains any points inside the Mandelbrot set, eventually the Z-plane square will surround the origin and get folded over itself at the next iteration.  So, in the former case you could interpolate across the square in the Z-plane, but because the Mandelbrot set boundary is so intricate, you need to pick smaller and smaller squares for it to work as you approach the boundary, so there's not much to gain from interpolation in the long run.

maybe this is obvious to mathematicians specialists, but it was new to me so:
wow! I just realize that what you call "strange results" is actually exactly what is happening: we see the jumpin of trajectories between the different period bulbs as depicted here:
https://de.wikipedia.org/wiki/Mandelbrot-Menge#Galerie_der_Iteration
As your starting value escape it jumps from bulb to bulb, your interpolation shows exactly that.