Say you have the IVP
$\dot{Z} = f(Z)$
where $Z: \mathbb{R} \to \mathbb{C}$, the function $f: \mathbb{C} \to \mathbb{C}$ contains complex coefficients, and $Z(0) = Z_{0}$ is complex. Will such an IVP always be equivalent to the IVP
$\dot{x} = Re(f(Z))$
$\dot{y} = Im(f(Z))$
with $x(0) + iy(0) = Z_{0}$ such that $Z(t) = x(t) + iy(t) \text{ for } t \geq 0$ ?
In other words, are you always able to rewrite an IVP with a complex ODE to an IVP of a system of ODEs? And if this is true, does it generalize to IVPs of systems of complex ODEs?