We investigate the effect of linguistic complexity on cognitive load in a dual-task scenario, namely simultaneous driving and language use. To this end, we designed an experiment where participants use a driving simulator while listening to spoken stimuli and answering comprehension questions. Online physiological measures of cognitive load, including the recently established Index of Cognitive Activity, as well as measures of performance in both tasks have been collected with high temporal resolution. The resulting aligned data streams can be used to test a vast array of different hypotheses about the relationship between performance, difficulty, and cognitive load in dual tasks at various levels of temporal resolution and linguistic structure. We present results of the data analysis, including evidence that different linguistic structures may cause measurable changes in cognitive workload on a very fine temporal scale in cases of increased primary task difficulty.