And in fact the Mandelbulb formula is completely wrong and incoherent in terms its algebra, but it produces beautiful images, which is in itself a sign that, perhaps, algebraic correctness is not what we always want.
Now it's possible to try to extend this geometric process to three dimensions regardless of any algebraic correctness. Instead of thinking in algebra and in the usual iteration formula w->w^2+c, this time we think on the classic 2D set as the result of a geometric process of iterating points by squaring their distance to the origin, rotating them by an angle equal to the current angle with the positive x axis, and then translating by c. While usually Quaternions, hypercomplex numbers and all other sort of (often inconsistent) algebras had been used in previous attempts, the construction of the Mandelbulb takes a different approach.
But, the Mandelbulb was one of those attempts. Unlike 4D Julia sets which naturally arise from simply generalizing the 2D complex Julia sets to quaternions, all efforts to produce an universal and intersting 4D Mandelbrot set been futile, and to this day (2020) nobody has found a satisfactory fractal. The idea of the Mandelbulb fractal came to be through yet another attempt at creating a 3D fractal that is as iconic as the traditional 2D Mandelbrot set is. A Mandelbulb fractal rendered in realtime over mobile phone footage