How much do you know about vectors? You can take two points and find the difference between them, as a vector; this is the analogue of the slope of a line in $\mathbb{R}^2$. If the vector is $\mathbf{v}$ and one point is $a$, then points on the line are of the form $a+\lambda\mathbf{v}$, where $\lambda$ is real. (Unless you're talking about a "complex line," which is really a plane.)
Also, if you're just uncomfortable with $\mathbb{C}^n$, you can just rewrite all your points as points in $\mathbb{R}^{2n}$, like $(Re(z_1),Im(z_1),\dotsc,Re(z_n),Im(z_n))$. This is an isomorphism of the real vector space structure of $\mathbb{C}^n$, so in particular it preserves lines and addition. The only thing you lose is complex multiplication.
EDIT: If you're looking for a complex line, just let $\lambda$ be complex in the above vector expression.