A large part of strong-field physics relies on trajectory-based semiclassical methods for the description of ionization and the subsequent dynamics, both for intuitive understanding and as quantitative models, including the workhorse Strong-Field Approximation and its various extensions. In this work I examine the underpinnings of this trajectory-based language in the saddle-point analysis of temporal integrals deformed over complex contours, and how this formalism can be extended to include interactions with the ion. I show that a first-principles approach requires the use of complex-valued positions as well as times, and that the evaluation of the ionic Coulomb potential at these complex-valued positions imposes new constraints on the allowed temporal integration contours. I show how the navigation of these constraints is infused with physical content and a rich geometry, and how the correct traversal of the resulting landscape has clear consequences on the photoelectron spectrum. <br/><br/> References:<br/> 1. E. Pisanty and M. Ivanov. Phys. Rev. A 93, 043408 (2016).<br/> 2. E. Pisanty and M. Ivanov. J. Phys. B: At. Mol. Opt. Phys. 49, 105601 (2016).