Generally speaking, the dot/scalar product is used to define the angle between 2 abstract vectors via $$u\cdot v=\vert u\vert \vert v\vert \cos\varphi$$ In the case of COMPLEX vector spaces, supposing complex scalar product we find the following problem to define the angle $\varphi$:
Example 1. $u=3+2i$, $v=-2+5i$. Then, define the quantity $$Q=u\cdot v=\vert u\vert \vert v\vert \cos\varphi= \sqrt{13}\sqrt{29}\cos\varphi=\overline{u}v=(3-2i)(-2+5i)=4+19i$$ Note that imposing the real part equal to the complex scalar product product projected onto the real line is a posible definition, BUT, it would also be a possible definition $\sqrt{\overline{Q}Q}$, wouldn't it? And so, we have the euclidean/hermitian angle solution. So, there is an ambiguity when we want to define the angle and the complex scalar product depending if we want a REAL-valued scalar product (euclidean angle follows), or we admit a complex-valued hermitian product (hermitian angle follows). Which one is better?
How to define the angle? Should we solve the equation for cosine IN THE COMPLEX REALM? Using the isomorphism con the real plane, one should be tempted to take the real part of $4+9i$, but is this true in higher dimensional complex spaces? Should be take the real part?
Example 2. Take the complex space $M_{2x2}(\mathbb{C})$. Take the scalar product between complex matrices to be defined by $A\cdot B=\mbox{tr}(A^+B)$.Define the matrices $\sigma_i$, $i=1,2,3$ as the Pauli matrices: $$ \sigma_1=\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix} $$ $$ \sigma_2=\begin{pmatrix} 0 & -i\\ i & 0\end{pmatrix} $$ $$ \sigma_3=\begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix} $$ Let C be the matrix: $$C=\begin{pmatrix} 3 & 2i\\ -5i & 4\end{pmatrix}$$ And now, for instance $$C\cdot \sigma_1=\mbox{tr}(C^+\sigma_1)=\mbox{tr}\left(\begin{pmatrix}3 & 5i\\ -2i & 4\end{pmatrix}\begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}\right)=\mbox{tr}\begin{pmatrix}5i & 3\\ 4 & -2i\end{pmatrix}=3i$$ $$C\cdot \sigma_2=\mbox{tr}(C^+\sigma_2)=\mbox{tr}\left(\begin{pmatrix}3 & 5i\\ -2i & 4\end{pmatrix}\begin{pmatrix} 0 & -i\\ i & 0\end{pmatrix}\right)=\mbox{tr}\begin{pmatrix}-5 & -3i\\ 4i & -2\end{pmatrix}=-7$$ The angle between $C,\sigma_2$ is easy to find out, since the scalar product was real. But, what happens with the angle between $C,\sigma_1$? Should we take the real part and consider orthogonal those 2 matrices? Or, shouldn't it be more natural if the dot product is complex valued to define the quantity below?: $Z=u\cdot v$ and $$\sqrt{\overline{Z}{Z}}=\vert u\vert \vert v\vert\cos\varphi$$.
Is all this meaning that there is NO a unique way to define the angle when complex (or even hypercomplex) numbers are involved?
After all the examples above, the question I wanted to be be answered is the following:
what is the best possible(more natural?) definition of scalar product/angle over a field (algebra/division algebra)? Is the real valued one or the K-valued version? Or should we accept that the 2 definitions are available?
