The following fractal is a Mandelbrot set with the quadratic equation replaced by powers, also known as a tetration escape fractal.

I generated this fractal thirty years ago. By extending tetration to the complex numbers I hope to plot the tetrational map also referred to as the pentation escape fractal.

Computing the Mandelbrot set is greatly accelerated by compiling using FunctionCompile. Unfortunately using pixel0^pixel instead of pixel^2+pixel0 in the quadratic equation breaks the compilation. I'm afraid that without a useful FunctionCompile statement I will not come close to the speed needed to plot a pentation fractal.

basicFun = Function[{Typed[pixel0, "ComplexReal64"]},
   Module[{iters = 1, maxIters = 10000, pixel = pixel0},
    While[iters < maxIters && Abs[pixel] < 10000,
     pixel = pixel0^pixel;
     iters++];
    iters]
   ];
compiledFun = FunctionCompile[basicFun]
ArrayPlot[
 Table[compiledFun[x + I y], {y, -2.5, 2.5, .0025}, {x, -3., 
   3., .0025}]]

What is the recommended method of compiling pixel0^pixel where both are complex values?