The wavefunction is ubiquitous in quantum mechanics, quantum chemistry and bosonic quantum information theory. For fermions, the zeros of the wavefunction are central in determining chemical properties, yet it is unclear how to extract the non-classical properties of a bosonic quantum system from its wavefunction. To tackle this problem, we study the zeros of the wavefunction of bosonic quantum systems. Our main technical result is that the wavefunction of most bosonic quantum systems can be extended to a holomorphic function over the complex plane, allowing us to leverage powerful results in complex analysis for the study of the wavefunction and its zeros. As a consequence, we prove a version of Hudson’s theorem for the wavefunction and we characterize the Gaussian dynamics of non-Gaussian quantum states by the motion of the zeros of their wavefunction. Our findings suggest that the non-Gaussianity of quantum optical states can be detected by measuring a single quadrature of the electromagnetic field, which we demonstrate in a companion paper [arXiv:2507.23005]. More generally, our results show that the non-Gaussian features of bosonic quantum systems are encoded in the zeros of their wavefunction.