There is at least one result saying that the Mandelbrot set is undecidable, and there might be more, but I think it (or they all) use real computation rather than Turing machines. This makes some sense, as $\mathbb{C}$ is connected, so the only decidable subsets of it are $\{\}$ and $\mathbb{C}$ itself. However, I've been reading about reverse mathematics, and I was wondering if the set is computable in the representations used. The Mandelbrot set is easily seen to be the complement of an effectively open set, and I'd guess asking whether it is separably closed would run into the open problem of are the hyperbolic components dense. This still leaves located closed, so the question follows:

Let $f : \mathbb{Q}[i] \to \mathbb{R} \hspace{.05 in}$ be given by $f(q) := \operatorname{min}(\{d(q,z) : z\in (\operatorname{Mandelbrot set})\})$. Is $f$ computable?