Conventional survival metrics, such as Harrell's concordance index (CI) and the Brier Score, rely on the independent censoring assumption for valid inference with right-censored data. However, in the presence of so-called dependent censoring, where the probability of censoring is related to the event of interest, these metrics can give biased estimates of the underlying model error. In this paper, we introduce three new evaluation metrics for survival analysis based on Archimedean copulas that can account for dependent censoring. We also develop a framework to generate realistic, semi-synthetic datasets with dependent censoring to facilitate the evaluation of the metrics. Our experiments in synthetic and semi-synthetic data demonstrate that the proposed metrics can provide more accurate estimates of the model error than conventional metrics under dependent censoring.