Physiological phenomena exhibit complex behaviours arising at multiple time scales. To investigate them, techniques derived from chaos theory were applied to physiological signals, providing promising results in distinguishing between healthy and pathological states. Fractal-like properties of electrodermal activity (EDA), a well-validated tool for monitoring the autonomic nervous system state, have been reported in previous literature. This study proposes the multiscale complexity index of electrodermal activity (MComEDA) to discern different autonomic responses based on EDA signals. This method builds upon our previously proposed algorithm, ComEDA, and it is empowered with a coarse-graining procedure to provide a view at multiple time scales of the EDA response. We tested MComEDA's performance on the EDA signals of two publicly available datasets, i.e., the Continuously Annotated Signals of Emotion (CASE) dataset and the Affect, Personality and Mood Research on Individuals and Groups (AMIGOS) dataset, both containing physiological data recorded from healthy participants during the view of ultra-short emotional video clips. Our results highlighted that the values of MComEDA were significantly different (p -value < 0.05 after Wilcoxon signed rank test with Bonferroni's correction) when comparing high- and low-arousal stimuli. Furthermore, MComEDA outperformed the single-scale approach in discriminating among different valence levels of high-arousal stimuli, e.g., showing significantly different values for scary and amusing stimuli (p -value = 0.024). These findings suggest that a multiscale approach to the nonlinear analysis of EDA signals can improve the information gathered on task-specific autonomic response, even when ultra-short time series are considered.