Let $M$ be the Mandelbrot set: there exists a unique series
$$
\psi(z) := z + \sum_{m=0}^{+\infty} b_m z^{-m} = z - \frac{1}{2} + \frac{1}{8} z^{-1} - \frac{1}{4} z^{-2} + \cdots
$$
which defines a conformal bijection between the complement $\{z\in\mathbb{C} : |z|>1\}$ of the closed unit disk in $\mathbb{C}$ and the complement of the Mandelbrot set. This function $\psi$ is sometimes known as the “Jungreis function”, see the answers to this question for more. The argument $\arg(\psi^{-1}(w))$ for a point $w\not\in M$ is called the **external angle** of $w$.

There exists an easy way to “almost” compute $\arg(\psi^{-1}(w))$, or $\psi^{-1}(w)$ itself for that matter: indeed, if we let $p_0(w) := w$ and $p_{i+1}(w) := p_i(w)^2 + w$, then $$ p_n(\psi(z)) = z^{2^n} + o(1) $$ as $z\to\infty$, so $\psi^{-1}(w)$ can be “almost” computed as the limit of the $(2^n)$-th root of $p_n(w)$ as $n\to+\infty$. The reason for the “almost” is that, while this indeed allows for computation of the modulus $|\psi^{-1}(w)|$, it leads to an indetermination between $2^n$ values on the argument. The formula $$ \psi^{-1}(w) = w \mskip3mu \prod_{n=1}^{+\infty} \left(1 + \frac{w}{p_{n-1}(w)^2}\right)^{1/2^n} $$ is no better (it also requires computing $(2^n)$-th roots and one cannot simply take the principal determination; my understanding is that one needs to find a determination of $\Big(1 + \frac{w}{p_{n-1}(w)^2}\Big)^{1/2^n}$ that is continuous outside of $M$, which seems computationally intractable).

So, **is there a way to lift this square root indeterminacy and compute external arguments** for arbitrary $w\not\in M$? Is there an algorithm that does this in a reasonably efficient way (which excludes, e.g., trying to trace external rays outwards towards infinity)?

I was unable to find anything relevant in the literature. There is a 1986 paper by Douady titled “Algorithms for computing angles in the Mandelbrot set” which seems promising, but it seems to concerns the computation for points of $M$, not points outside $M$. This web page about the *Mandel* program actually discusses the issue (in the section called “Computation of the external argument”), but the description is vague (e.g., where it speaks of a “modified” function $\arg(z/(z-c))$), and the conclusion that “the discontinuities are moved closer to the Mandelbrot set” is not too promising.