Analysis of the boundary of the Mandelbrot set Motivation: The Mandlebrot set is a simply connected set with an infinitely complex boundary, but CAN one move from interior to the exterior of this topological space by just crossing over a finite set of points?
Application: I am modeling the leakage of emission of EM waves, through a fractal forest where dispersion occurs every time an edge of the forest is crossed
Let $\mathbb{M}\subset \mathbb{C}$ be the Mandelbrot Set. Let $\partial \mathbb{M}\subset \mathbb{M}$ be its boundary. Consider two points $\mathbb{z_1}\in \mathbb {M}$ and $\mathbb{z_2}\in\mathbb{C}\notin \mathbb{M}$
My question is: 


*

*does there exist a path $\mathbb{P}$ $ \subset \mathbb{C}$ with initial point $\mathbb{z_1}$ and terminal point $\mathbb{z_2}$? 

*Is $\mathbb{P}\bigcap\partial\mathbb{M}$ a finite set?
 A: Your notation is unusual, and I am not sure whether I entirely understand it.
I shall take your question to mean: Can every point of the Mandelbrot $M$ set be connected to $\infty$ by a path that intersects the boundary $\partial M$ in only finitely many points?
This is connected to a very famous conjecture, namely that The Mandelbrot set is locally connected. (See The deep significance of the question of the Mandelbrot set's local connectedness?.)
Indeed, if the Mandelbrot set is locally connected, then - in particular - every point of the boundary of M is accessible from $\infty$. Note that "finitely many points" can be replaced by "one point". (Recall that the Mandelbrot set is full, and hence every interior component is simply connected.)
If the Mandelbrot set is not locally connected, then it follows e.g. from the theory of "fibers", as formulated by Dierk Schleicher, that there exists a point of the boundary that is not accessible from the complement, and hence the answer to your question would be negative.
(Caution. This statement is not true for general compact sets: a compact and full connected set can have every point accessible, but not be locally connected. The statement above uses the specific structure we know about the Mandelbrot set.)
If you start asking for curves with specific geometric properties, the question will become more subtle. For example, if you ask for smooth curves, the answer is 'no' in general (as the Mandelbrot set spirals at many points).
