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 ?