You can solve the equation shown by posing it as a problem mapping from R2->R (rather than the complex numbers C->R). You can do this by expanding it out if you substitute the term a+bi for z.
Then I used three different methods of finding minima: Newton's method, fixed step-size gradient method, and the conjugate gradient method. The gradient methods were sloooooow especially at the center where the gradient is small.
Yellow means it converged to the point [a,b] = [0,1]. Green: [1,0]. Blue: [0,-1]. Red: [-1,0]. Orange converges to the origin [0,0]. White means it didn't converge to any of those but didn't go wildly careening off in some direction (diverge). Black means it did diverge, though I actually came up with that late in the game so some of these plots might not be totally accurate according to that legend.
In other news, some progress on the laser cutter but that's about it.