How to find random points of interest inside the Mandelbrot Set

[kronikel]

I am seeking a way to generate a point and a depth to zoom into that point at random that will always, or at least close to always, show some sort of detail.
We would want to stay away from points in the cardioid and the bulbs, and of course anything with a distance greater than 2 from the origin. This isn't so hard to accomplish, but there are an infinite number of places still left that will either be a part of the set or just return one solid number of iterations over the entire image.
The ideal algorithm for what I am doing will return a diverse range of different kinds of patterns in the set. It does not need to give points that are all over the set though, in fact an easy way to do it may be to find one point and simply choose random depths to zoom into that point. Does such a point exist that can be zoomed into (using a finite but very large range of different depths) will always satisfy two conditions - a wide variety of different patterns will be observed, and never returning an array of all the same (or close to the same) number of iterations?
If there isn't a known way to do something like this, my backup plan is to generate random parts of the set and have an algorithm that checks how many different values are present in the image and rejects any sum of different values below a certain threshold, or have different points selected by hand and then find out how far you can zoom into each point before you hit a blank wall, then use this depth as the range for the random depth chosen at this given point. But either of these is of course inefficient.

[Adam Majewski]

http://mathr.co.uk/blog/2016-03-05_julia_morphing_symmetry.html
http://mathr.co.uk/blog/2016-02-25_automated_julia_morphing.html

HTH