\begin{equation} f(x) = e^{ix} = \cos (x) + i\sin (x) \end{equation}
this brings a circle of radius one, because in every point, velocity is orthogonal to where you are (because f’(x) = if(x), and multiplying by i accounts for a rotation of 90 degrees. And so,
\begin{equation} z = re^{i\theta} \end{equation}
gives any point in the imaginary polar plane.