codimension of the zero set of a polynomial in several variables and their conjugates
Anything can happen. Writing $z_j=x_j+iy_j$ for real $x_j$ and $y_j$, we can express $x_j$ and $y_j$ as polynomials in $z_j$ and $\overline{z_j}$: $x_j=\frac{z_j+\overline{z_j}}{2}$ and $x_y=\frac{z_j-\overline{z_j}}{2i}$. From separating real and imaginary parts, this means we can get up to two real polynomials in the $x_j$ and $y_j$. But this means we can get any dimension of vanishing set we want: consider the polynomial $$(\sum_{j=1}^r x_j^2)+(\sum_{j=1}^s y_j^2)$$ which cuts out a subset of codimension $r+s$.