Let $p\in\Bbb C^n$ (with $n\ge2$) be an attractive fixed point for $F\in\operatorname{Aut}\Bbb C^n$ (i.e. holomorphic bijection), that is $F(p)=p$ and all the eigenvalues of $F'(p)$ are in modulus $<1$.
The basin of attraction of $F$ in $p$ is defined as $$ \Omega_{F,p}:=\{z\in\Bbb C^n\;:\;\lim_{j\to+\infty}F^{(j)}(z)=p\} $$ where $F^{(j)}$ is the composition of $F$ with itself $j$-times.
Is $\Omega_{F,p}$ starshaped with respect to $p$?
The answer is no, although it doesn't seem so easy to give a rigorous counter-example.
First, it is not the case that immediate basins of attracting fixed points are star-like for quadratic polynomials. Proof by picture: https://upload.wikimedia.org/wikipedia/commons/c/c3/Julia_set_of_the_quadratic_polynomial_f%28z%29_%3D_z%5E2_-1.12%2B_0.222i.png
Then given say the quadratic polynomial $f_c(z)=z^2+c$ in the picture, you can perturb it to get a Henon map $g_c(z,w)=(f_c(z)+\epsilon w,z)$ that is an automorphism of $\mathbb C^2$. As $\epsilon \to 0$, the attracting basin of $g_c$ degenerates to that of $f_c$, so I expect it not to be star-like.