The increasing use of clustering in socially sensitive domains has raised concerns about fairness. While several fairness-aware clustering methods have been proposed, a key open question remains: what is the cost of enforcing fairness in clustering? In this work, we address this question by introducing two classes of fairness cost measures. First, we define similarity-based costs that quantify how much a fair clustering diverges from its unfair baseline, using extensions of Normalized Mutual Information (group NMI) and cluster alignment metrics. Second, we introduce novel counterfactual-based costs that capture the minimal feature changes required for individuals to move from their original to their fair cluster assignments. This counterfactual framework also enables feature-level analysis, identifying which features contribute most to fairness interventions. We apply our approach on four real-world datasets using two fairness criteria, namely balance fairness and social fairness. Our results show that social fairness tends to preserve the original clustering structure better than balance fairness, although it does not always yield lower individual counterfactual costs. Furthermore, our analysis reveals implicit biases, with features such as marital status emerging as proxies for sensitive attributes.