Obviously, only the points in the boundary $p\in\partial M$ are interesting. I managed to prove a few examples:
For $p=\frac 14$ the set remains connected, since $M\cap \mathbb R=[-2,\frac 14]$ and the Mandelbrot contains the main cardioid.
For $p=-\frac 34$ the set loses connectivity since for a $c=-\frac 34+\epsilon i$ the fixed point which is parabolic at $\epsilon=0$ is $z_0=\frac 12-\sqrt{1-\epsilon i}$ and we have $|f'(z_0)|=1+\frac{\epsilon^2}4+O(\epsilon^4)$ so it is repelling for small $\epsilon$.
For $p=-2$, consider a parameter $|c|=2$, and we have $f^2(0)=c^2+c$ so $|f^2(0)|=2|c-1|$ which is only $\leq 2$ when $c=-2$ which means $M\setminus\{-2\}$ is contained in the open disk $B_2(0)$. Using a rather contrived argument I was able to prove that when $c\to -2$ with argument bounded by a certain positive function, the periodic point near $-1$ is attracting and attracts the origin so $c\in M$. Thus $M\setminus\{-2\}$ is connected.
Is there any relationship between this property and the other properties of points on the boundary of the MAndelbrot set? For example whether they are parabolic or not?
