Let $X$ be an affine variety $\operatorname{Spec} \mathbb C[x_1,\dots, x_n]/(f_1,\dots, f_m)$. Suppose the set of closed points gives a smooth complex analytic variety in $\mathbb C^n$. Pick any $p\in X$, and let $U$ be a small open ball in $\mathbb C^n$.
Question: Is $U\cap X$ Zariski dense in $X$?
I think this is correct but would like more careful thinking. A baby example may be as follows: the open unit disk in $\mathbb C$ is Zariski dense.
Let $Z \subset X$ be the Zariski closure of $U \cap X$. Then $Z$ is the vanishing locus of functions $g_1, \dotsc, g_s \in \mathbb C[x_1, \dotsc, x_n] / (f_1, \dotsc, f_m)$. Since the $g_i$ vanish in the neighbourhood $U \cap X$ of $p$, they vanish identically on the full connected component of $X$ by the identity theorem for holomorphic functions.