geometric interpretation of quadratic equation with complex coefficients

When an equation has real coefficients and non-negative discriminant, the geometric meaning of it's roots is intersection of the function with the x-axis.

I know how to get roots of quadratic equation with complex coefficients, I just wonder if there is any geometric interpretation for that.

-

Try my paper, Graphing the Complex Roots of Quadratic Functions on a Three Dimensional Coordinate Space, detailing a geometric construction detailing how to derive the complex roots geometrically.

-
Say you have a polynomial $F(x_1,...,x_n)$ with real or complex coefficient. Then you can consider it as definining a function $$K^n\longrightarrow K\qquad\qquad(*)$$ where $K$ is either $\Bbb R$ or $\Bbb C$. Simply, the value associated to an $n$-ple $\vec a=(a_1,...,a_n)$ is $F$ evaluated at $\vec a$. In geometric terms, solving the equation $$F(x_1,...,x_n)=0$$ means finding the zero-locus of the function $(*)$ which is some subset $Z_F\subseteq K^n$. The case you are familiar with is that of $n=1$, where $Z_F$ is typically a set of isolated real or complex points.
Studying the geometry of the sets $Z_F$ is the foundational motivation of Algebraic Geometry