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For this study, the effects of thermal annealing and compressive strain rate on the complexity of the serration behavior in a Zr-based bulk metallic glass (BMG) was investigated. Here, as-cast and thermally-annealed (300 °C, 1 week) Zr52.5Cu17.9Ni14.6Al10Ti5 BMG underwent room-temperature compression tests in the unconstrained condition at strain rates of 2 × 10−5 s−1 and 2 × 10−4 s−1. The complexity of the serrated flow was determined, using the refined composite multiscale entropy technique. Nanoindentation testing and X-ray diffraction characterization were performed to assess the changes in the microstructure and mechanical properties of the BMG that occurred during annealing. The results indicated that the BMG did not crystallize during annealing in the prescribed heating condition. Nanoindentation tests revealed that annealing led to a significant increase in the depth-dependent nanoindentation hardness and Young’s modulus, which were attributed to the structural relaxation in the glass. Furthermore, both annealing and an increased strain rate resulted in a marked enhancement in the complexity of the serrated flow during compression. It was concluded that the increase in the sample entropy with increasing strain rate is related to an increase in the number of defect interactions during the serrated flow.