I have quantity (say $V$) represented in (x,y,z) coordinate, each has complex (phase and amplitude) value, i.e., $$ V = |V_x| \exp(j \gamma_{x}) a_x + |V_y| ~\exp(j \gamma_{y}) a_y + |V_z| ~\exp(j \gamma_{z}) a_z$$
where $|V_x|$ and $\gamma_{x}$ are the amplitude and phase of $V$ in the unit direction $a_x$ (i.e., in x), same for $a_y$ and $a_z$. I need to convert that quantity to spherical coordinates $(r,\theta,\phi)$. I know how to do that if we have only real components, but not sure how to proceed when we have real and imaginary for each component.
In particular, if $\gamma = 0$ for all directions, then we have $$V_r = \sin(\theta) \cos(\phi) V_x + \sin(\theta) \sin(\phi) V_y + \cos(\theta) V_z $$ $$V_\theta = -\cos(\theta) \cos(\phi) V_x + \cos(\theta) \sin(\phi) V_y - \sin(\theta) V_z $$ $$V_\phi = -\sin(\phi) V_x + \cos(\phi) V_y $$