Geometric intuition of perpendicular complex vectors

I want to grasp the idea of orthogonality and angle between complex vectors. Some things are very confusing to me like if i take a vector in $mathbf{C}^1$ (say $a+iota b$) and it’s conjugate $a – iota b$ then the the dot product is always $> 0$ and using the standard equation for angle can claim it can never be orthogonal.

However my mind immediately tries to think $1+ iota$ and $1- iota$ in complex plane and how they are perpendicular in the complex plane geometrically.

So what is the right way to view complex vectors geometrically ?