Conformal maps from the complement of the Mandelbrot set to the disk are well known. Also, it is known that there exists a surjective conformal map from the interior of the Mandelbrot set to the disk by the Riemann mapping theorem.
Is an explicit example of a conformal map from the Mandelbrot set to the unit disk known?
For the revised question suggested in the comment:
You can use the derivative of the limit cycle of points in hyperbolic components as a coordinate in the unit disc. I believe the mapping is conformal, but I don't have a proof handy...
Practically, you would check candidate periods $p$ where $|z_p|$ reaches a new minimum in iterations of $z_{n + 1} = f_c(z_n)$. Use Newton's method to solve $z = f_c^p(z)$, using $z_p$ as initial guess. If $\left|\frac{\partial}{\partial z} f_c^p(z)\right| \le 1$ (evaluated at the root you found), then $c$ is inside a hyperbolic component of period $p$, and $w = \frac{\partial}{\partial z} f_c^p(z)$ is mapped to the unit disc. I have no proof of correctness of this algorithm, but it seems to work...