We are interested in the boundary of the Mandelbrot set. It is closed, so does have a boundary. It seems a good idea to increase the boundary by making holes in the set. Which is easy to do. We then have not lost anything and have gained additional boundary.
The Mandelbrot set is defined by repeated application of the function $ f(z) = z^2 + c $ starting from z=0. The set is then those c for which z does not tend to infinity. Of these about a third (mostly near the origin) tend to a point, and some cycle. For example if c=-2. z=0 goes to -2, then to 2 and stays at 2. If c = i, z cycles between -1 + i and -i.
We can make holes by excluding c for which z either goes to a point or to a cycle. But I am not able to see in any detail what the resulting Mandelbrot set looks like. So this question is not theoretical. It is a request for a diagram so I can see what the Mandelbrot set looks like with these holes. I am hoping that this is sufficiently interesting to make it worthwhile for someone to produce this diagram.
There is a similar situation on the inside to the outside where we ask does z go to infinity? For any c we ask whether z has stopped moving or is cycling. We can give values of c different colours for z cycles of different lengths. There are certainly enough cycles, at least of length 2, to make this worthwhile. If a point c results in z cycling, a point nearby is also likely to result in z cycling with the same cycle length. This implies we have pools of colours.
I am putting some findings in my comments about the answer to this question.