As a general rule, since the "length" in standard geometry is the square root of a computed value, and we'd like "length" to be well-defined, we don't allow complex lengths.
There is a neat way to generalize geometry on complex numbers. If $x=(x_1,x_2),y=(y_1,y_2)\in\mathbb R^2$, then we define $x\cdot y = x_1y_1+x_2y_2$. Then $\sqrt{x\cdot x}$ is the distance from $0$ to $x$.
When $x,y\in\mathbb C^2$, we change our definition of the dot product:
$x\cdot y = x_1\overline{y_1} + x_2\overline{y_2}$
Where, for $z\in\mathbb C$, $\overline z$ is the complex conjugate.
Then $y\cdot x = \overline{x\cdot y}$.
In particular, $x\cdot x$ is now again a positive real number, so we can take its positive square root to get the distance of $x$ from zero, $|x|=\sqrt{x\cdot x}$, and we have $|x|=0$ if and only if $x=0$. This distance happens to be exactly the same distance that we'd get if we considered $\mathbb C^2$ to be $\mathbb R^4$.
Another thing you lose in complex geometry is the notion of "between-ness." Given three distinct points on a real line, there is exactly one of them that is between the other two. There is no such notion of "between" for points of a complex line. Essentially, the complex line is not "linear."