It is not hard to see that the Gaussian integers $\mathbb{Z}[i]$ are Zariski-dense inside $\mathbb{C}$, seen as an affine space over $\mathbb{C}$. Consider now the set $$D = \{(z,\overline{z}) \in \mathbb{C}^2\colon z \in \mathbb{Z}[i]\}\,.$$ It is no longer clear to me if $D$ is Zariski-dense inside the $\mathbb{C}$-affine space $\mathbb{C}^2$.
The diagonal clearly is not, as it is the zero set of $f(x,y) = x-y$. However, it seems to me that if polynomials could isolate the set $D$ from the whole $\mathbb{C}^2$, then "complex conjugation" should be expressible as a polynomial function, which most certainly it is not. But I couldn't figure out a proof of this either.