Several recent proposals to embed inflation into high-energy physics rely on inflationary dynamics characterized by a strongly non-geodesic motion. This in turn relaxes the conditions of slow-roll to allow for potentials that are steep in Planck units, a welcome feature in view of the eta problem and the recently much discussed swampland conjectures. In this talk I will present a general framework to study non-Gaussianities in these type of models. This consists in deriving a non-standard single-field effective field theory with imaginary speed of sound supported by first-principle numerical computations. I will then highlight how non-geodesic trajectories in fields space naturally leads to “hyper non-Gaussianities” and a possible loss of perturbative control. This allow us to derive model-independent constraints which sharpen the range of allowed theoretical constructions.