Is $\partial M_d$ continuously determined by $d$?

Question

This question is inspired by a question on math.stackexchange:

https://math.stackexchange.com/questions/1707291/is-the-generalized-mandelbrot-set-a-fractal-in-the-d-dimension/2575089

The animation in that question looks very "smooth" and I wonder:

Is the set of unstable parameters $c\in\mathbb{C}$, of $z\to z^d + c$, continuously determined by $d$ (for $d$ in some suitable real interval)?

This is interesting to me because if the answer is "no" then there ought to be a kind of "meta-bifurcation-locus" in $\mathbb{C}\times\mathbb{R}$, and I wonder if that set has been studied, named, visualized, etc.