Complex star-shaped and convex domains are the published Euclidean notions under the identification $\mathbb{C}={\mathbb{R}}^2$

We use the identification $\mathbb{C}={\mathbb{R}}^2$ from $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves without changing its Euclidean topology or line segments. Thus an open set $U\subseteq \mathbb{C}$ is star-shaped with respect to $a\in U$ precisely when the whole segment $(1-t)a+tz$, $0\le t\le 1$, lies in $U$ for every $z\in U$, as in Star-shaped open subsets of Euclidean space. It is convex precisely when the segment $(1-t)z+tw$ lies in $U$ for every $z,w\in U$, as in A convex subset of ${\mathbb{R}}^m$ contains every line segment between two of its points.

A complex domain is nonempty, open, and connected by A complex domain is a nonempty connected open subset of $\mathbb{C}$. Consequently, a convex complex domain is star-shaped with respect to each of its points: after fixing $a\in U$, convexity applied to $a$ and any $z\in U$ gives the required segment. Connectedness is not needed for that implication.