The goal of voice anonymization is to modify an audio such that the true identity of its speaker is hidden. Research on this task is typically limited to the same English read speech datasets, thus the efficacy of current methods for other types of speech data remains unknown. In this paper, we present the first investigation of voice anonymization for the multilingual phenomenon of code-switching speech. We prepare two corpora for this task and propose adaptations to a multilingual anonymization model to make it applicable for code-switching speech. By testing the anonymization performance of this and two language-independent methods on the datasets, we find that only the multilingual system performs well in terms of privacy and utility preservation. Furthermore, we observe challenges in performing utility evaluations on this data because of its spontaneous character and the limited code-switching support by the multilingual speech recognition model.