In a stationary and axisymmetric spacetime, there are two Killing vectors, say $\zeta^\mu$ and $\xi^\mu$, one timelike and one space like.
I understand that for a real scalar field, $\phi$, that obeys the symmetry of this spacetime one can say
$$\zeta^\mu \nabla_\mu \phi = \xi^\mu \nabla_\mu \phi = 0 $$
and therefore since the gradient of $\phi$ is orthogonal to both Killing vectors, $\nabla_\mu \phi \nabla^\mu \phi \geq 0$.
Now, my question is in regards to how this logic translates across if $\phi$ becomes a complex scalar field, where $\Phi = \phi_1 + i\phi_2$.
$\nabla_\mu \Phi \nabla^\mu \Phi \geq 0$ no longer makes sense to me, since the gradient is complex, however is it a valid comment to say that the real components of the complex field must be spacelike or null, if the complex field is to still obey the spacetime symmetries, i.e. $\nabla_\mu \phi_i \nabla^\mu \phi_i \geq 0$