So there are some inconsistencies for which I don't understand about complex vector spaces in general.
We know the inner product space of a complex vector field has the property $<x,y> = \overline{<y,x>}$. So let $x = a + \alpha i$ and $y = b + \beta i$. We have $$<x,y> = (a + \alpha i)(b + \beta i) = ab + a\beta i + \alpha b i - \alpha\beta = (ab - \alpha\beta) + (a\beta + \alpha b)i$$but $$\overline{<y,x>} = \overline{(b + \beta i)(a + \alpha i)} = \overline{(ab - \alpha\beta) + (a\beta + \alpha b)i} = (ab - \alpha\beta) - (a\beta + \alpha b)i$$So $(ab - \alpha\beta) + (a\beta + \alpha b)i = (ab - \alpha\beta) - (a\beta + \alpha b)i$ and essentially $m + ni = m - ni$? How is this consistent? And why does the position of $x$ and $y$ need to be switched in the inner product defintion? i.e. wouldn't $<x,y> = \overline{<x,y>}$ suffice?