How to find the chosen parameter point

[gamma]

To draw Julia sets we need a point (to serve as parameter) in plane within reasonable distance from zero. Mandelbrot set can serve as a colorful guide to show us which points should have interesting results, such as the ones on the edge of the "main lake" between connected and disconnected Julia sets. Lets call this point a complex number C.

I usually notice that if I first zoom into Mandelbrot and then choose the point C, the resulting Julia set looks like that magnified area of Mandelbrot where I was searching for parameter of choice C. (For example, try using XaoS). What struck me as an unknown recently is that C points become forgotten, lost or even irrelevant in the process. To be precise, I don't see it per se as a dot on the screen. Surely C is saved in fractal software as a parameter for each image, that part is easy.

My question is which fractal would show us at least approximately where C is within the final Julia-like set? For example, an old screensaver in linux used inverse-julia set to morph endlessly and at all those times the program would show the point C walking around the screen. I can not say if there is a fundamental reason why the C would appear or if someone thought it would be fun to add it over the picture of inverse-julia. Perceptually, the C creates impression of a balanced relation and it's motion inertia and weight in that screensaver.

I presume that a julia-like set which is structured (convoluted, bent, wrapped) around point C would not be appropriately symmetric and maybe would not fit into definition of Julia set. My thoughts for today.