Note:
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In this question, a complex number is counted as a vector initiated from the origin.


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> Is there a holomorphic function $B:\mathbb{C}^2 \to \mathbb{C}$ such that for every two non zero complex numbers $z,w$ with $z/w \notin \mathbb{R},$ the vector $B(z,w)$ is a non zero vector indicating to the direction of the bisector of the angle $\angle (z,w) $?
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Motivation:
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The initial formula for the "Bisector" of $\angle (z,w) $ is $B'(z,w)=|z|w+|w|z$. But it is not a holomorphic function.(It is not even smooth at $z=0$ or $w=0$). So we search for a holomorphic remedy, a holomorphic function $B$ defined on whole $\mathbb{C}^2$ such that $B(z,w)$ is real proportional to $(|z|w+|w|z)$ via a non constant real function $\lambda$.


What about if we require that such $\lambda $ be positive(Non negative)?



Can the "Bisector" be represented by a holomorphic function?
Here is a better answer than the other answer I gave, which is currently accepted. It also answers some of your questions in the comments on that answer. Pick a branch of log, then $B$ and $ (zw)^{1/2}$ are holomorphic and their arguments differ by a multiple of $\pi$ wherever they are both defined. So their quotient is a real holomorphic function and hence a real constant wherever it is defined. So locally, $B$ has to be a real multiple of some branch of $ (zw)^{1/2}$.


