So I've been wondering about geometry on the complex plane. Points on this plane are denoted by $(a+bi)$, right? If we have the point $(1+i)$, the horizontal distance from the complex axis is $1$, and the vertical distance from the real axis is $i$. So the distance from the origin by Pythagorean theorem would be: $$ 1^2 + i^2$$ $$=1+(-1)$$ $$= 0?$$ But that would mean the point $(1+i)$ is on the origin which it it obviously not! Is there an explanation or is regular geometry like Pythagorean theorem not applicable to the complex plane.
As already mentioned in the comments, distance functions are typically real valued, not complex valued.
A distance function on a set $X$ is a map $X\times X\to \mathbb R$ that satisfies certain properties. Two of the properties include inequalities.
If you try to replace $\mathbb R$ with $\mathbb C$, this stops making sense because there is no ordering for $\mathbb C$ for those axioms to work with, at least, not one that is geometrically useful. Instead, $\mathbb C$ gets its metric from its underlying metric space $\mathbb R\times\mathbb R$.