Let $Rat_d$ be the set of all rational fraction of degree $d$ and $X_d \subset Rat_d$ be the bifurcation locus of rational fractions of degree $d$, i.e. the closure of the set of discontinuity of the map $f \mapsto J(f)$ defined on $Rat_d$, and associating to $f$ its Julia set (with the topology of Hausdorff distance).

Now denote by $M_d$ the quotient of $Rat_d$ under conjugation by homographies, and by $Y_d$ the projection of $X_d$ onto $M_d$. (if you consider polynomials only instead of rational fractions, then $M_2=\mathbb{C}$ and $Y_2$ is the boundary of the Mandelbrot set).

A result of Milnor gives an explicit diffeomorphism from $M_2$ to $\mathbb{C}^2$.

Question : is there somewhere a computer picture of $Y_2$ in the coordinates defined by Milnor ? (like a projection, or a family of cuts). Judging by the aspect of the Mandelbrot's set boundary, one would expect quite a complicated and interesting set.