Clinical machine learning is increasingly multimodal, collected in both structured tabular formats and unstructured forms such as free text. We propose a novel task of exploring fairness on a multimodal clinical dataset, adopting equalized odds for the downstream medical prediction tasks. To this end, we investigate a modality-agnostic fairness algorithm - equalized odds post processing - and compare it to a text-specific fairness algorithm: debiased clinical word embeddings. Despite the fact that debiased word embeddings do not explicitly address equalized odds of protected groups, we show that a text-specific approach to fairness may simultaneously achieve a good balance of performance classical notions of fairness. Our work opens the door for future work at the critical intersection of clinical NLP and fairness.