Apologies, don't have time to do a proper job of it right now, so here's a quick summary: Complex addition is basic Cartesian vector addition in the complex plane; complex multiplication is combined polar-coordinate rotation/scaling, by adding the two angles and multiplying the magnitudes. The Mandelbrot iteration formula is Z*Z+C, so there are only three transformations: double the angle, apply the scaling factor, and move the point by a fixed offset.
For each point in the graph, fix the distance between it and the origin: that's C. Center a wheel of some sort at the origin, find the angular distance from the point to the positive real line, then rotate the wheel by that amount. The magnitude scaling is the trickiest part, since you're moving it either toward or away from the origin depending on whether it's greater or less than one, and the scaling factor is proportional to the magnitude itself; but I'm pretty sure there's some sort of multiplicative transformation that could be done with an appropriately-placed lever.
So, take the aforementioned contraption, rotate the wheel, adjust the distance, then shift the point by the original offset. Repeat 70 times, or until the distance exceeds the machine's maximum. Mark the point appropriately, and pick a new one. The accuracy would be dubious near the boundary, of course, but it beats doing arithmetic...
Maybe one of these days I'll draw up usable schematics for how it would work. Mechanical computation has a certain charm to it.
