An optical field is described by the amplitude and phase, and thus has a complex representation described in the complex plane. However, because the only thing we can measure is the amplitude of the complex field on the real axis when not introducing an additional imaging system, it is difficult to identify how the complex field behaves throughout the complex plane. In this study, we interpret quantitative phase imaging methods via the Hilbert transform in terms of analytic continuation, manifesting the behavior in the whole complex plane. Using Rouche's theorem, we prove the imaging conditions imposed by Kramers-Kronig holographic imaging. The deviation from Kramers-Kronig holography conditions is examined using computational images and experimental data. We believe that this study provides a clue for holographic imaging using the holomorphic characteristics of a complex optical field.