Ketamine is an effective intervention for treatment-resistant depression (TRD), including late-in-life (LL-TRD). The proposed mechanism of antidepressant effects of ketamine is a glutamatergic surge, which can be measured by electroencephalogram (EEG) gamma oscillations. Yet, non-linear EEG biomarkers of ketamine effects such as neural complexity are needed to capture broader systemic effects, represent the level of organization of synaptic communication, and elucidate mechanisms of action for treatment responders. In a secondary analysis of a randomized control trial, we investigated two EEG neural complexity markers (Lempel-Ziv complexity [LZC] and multiscale entropy [MSE]) of rapid (baseline to 240 min) and post-rapid ketamine (24 h and 7 days) effects after one 40-min infusion of IV ketamine or midazolam (active control) in 33 military veterans with LL-TRD. We also studied the relationship between complexity and Montgomery-Åsberg Depression Rating Scale score change at 7 days post-infusion. We found that LZC and MSE both increased 30 min post-infusion, with effects not localized to a single timescale for MSE. Post-rapid effects of reduced complexity with ketamine were observed for MSE. No relationship was observed between complexity and reduction in depressive symptoms. Our findings support the hypothesis that a single sub-anesthetic ketamine infusion has time-varying effects on system-wide contributions to the evoked glutamatergic surge in LL-TRD. Further, changes to complexity were observable outside the time-window previously shown for effects on gamma oscillations. These preliminary results have clinical implications in providing a functional marker of ketamine that is non-linear, amplitude-independent, and represents larger dynamic properties, providing strong advantages over linear measures in highlighting ketamine’s effects.