In this paper, we study the complex bounded component analysis (BCA) problem, under the novel scenario that the magnitudes of the latent sources are bounded. This is opposed to the existing models that assume separate bounds on the real and imaginary parts, in which case the problem can be transformed into a real BCA problem, but loses the point of introducing complex numbers into the model in the first place. Unlike in the real case, it is hard to visualize a geometric interpretation for the complex BCA model. Nevertheless, we draw algebraic insights from prior work and propose a new formulation that uses the determinant of the mixing matrix as the identification criterion, and show that complex BCA is identifiable if the disked hull of the sources is sufficiently scattered in the complex hypercube. This result significantly extends the prior knowledge on BCA, and in a broader sense is perhaps the first identifiable unmixing model with parts of the sources being quadratically dependent (since the magnitude of the complex sources are bounded). We also present a new learning algorithm to solve the proposed complex BCA formulation based on linearized ADMM, and show numerically that the performance is surprisingly effective.