We present an algorithm for clustering complex-valued unit length vectors on the unit hypersphere, which we call complex spherical k-mode clustering, as it can be viewed as a generalization of the spherical k-means algorithm to normalized complex-valued vectors. We show how the proposed algorithm can be derived from the Expectation Maximization algorithm for complex Watson mixture models and prove its applicability in a blind speech separation (BSS) task with real-world room impulse response measurements. It turns out that the proposed spherical k-mode algorithm is on par with other state-of-the-art BSS algorithms in terms of signal-to-inference ratio gains although being far easier to implement and using fewer calculations.