pasmith said:
"Orthogonal" is not some absolute concept; it has meaning only in relation to a specific inner product.
Or phrased a bit more basically: Orthogonality depends on how angles are measured.
Whereas angles in a real Euclidean plane are usually measured by things like
that corresponds to an inner product defined by the identity matrix ##\boldsymbol M =\boldsymbol I ## (cp.
https://en.wikipedia.org/wiki/Inner_product_space#Euclidean_vector_space). A (squared##^*)##) length in the real plane is defined by ##x^2+y^2,## the inner product of a vector by itself. This is always a positive number as we would expect. But it is no longer automatically positive if the coordinates ##x,y## are allowed to be complex. The identity matrix is therefore no longer suited to measure angles and lengths. The "geodreieck" isn't an appropriate tool any longer. Besides, it would be difficult to use it if we all of a sudden have four real coordinates ##a,b,u,v## from ##x=a+i b\, , \,y=u+ i v.## However,
$$xx^\dagger =(a+ib)\cdot \overline{(a+ib)}=(a+ib)\cdot (a-ib)=a^2-i^2b^2=a^2+b^2$$
turns out to be always a positive real number (except for the zero vector, of course) and is thus suited to define a (squared##^*)##) length again. Hence taking the complex conjugate in the second ##^{**})## coordinate resolves the dilemma. This defines a complex version of an inner product, and via
$$
\cos \sphericalangle \left(\vec{x},\vec{y}\right)=\dfrac{\bigl\langle \vec{x}\, , \,\vec{y} \bigr\rangle }{\sqrt{\bigl\langle \vec{x}\, , \,\vec{x} \bigr\rangle\, \cdot \,\bigl\langle \vec{y}\, , \,\vec{y} \bigr\rangle}}=\dfrac{\vec{x}\cdot \vec{y}^\dagger}{\sqrt{\left(\vec{x}\cdot \vec{x}^\dagger\right)\cdot\left( \vec{y}\cdot \vec{y}^\dagger\right)}} \, , \,
$$
an angle defined by that inner product, i.e. the definition of how we measure lengths.
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##^*)## Taking the root of the formulas ##x^2+y^2## or ##a^2+b^2## is necessary by Pythagoras, or simpler: to make sure that the units like inches still fit. We get square inches from the formula, i.e. a squared length.
##^{**})## Choosing the first coordinate is equally possible, as long as it is always the first one, or always the second one. Physicists and mathematicians use standardly the opposite coordinate for conjugation. IIRC then physicists use the first argument for conjugation and mathematicians the second. Don't ask me why or if I am even sure. It's reasonable to check it on the case, author by author.