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In this study, a novel sparsity-based space-time adaptive processing algorithm based on the complex-valued Homotopy technique is proposed for airborne radar applications. The proposed algorithm firstly extends the existing standard real-valued Homotopy method to a more general complex-valued application using the gradient approaches. By exploiting the sparsity of the clutter spectrum in the whole spatiotemporal plane, the proposed algorithm recovers the clutter spectrum via the proposed complex Homotopy algorithm and then uses it to estimate the clutter covariance matrix, followed by the spacetime filtering and the target detection. Furthermore, the implementations of the proposed algorithm are detailed. The computational complexity analysis shows that the proposed algorithm has a lower-computational complexity than the existing complex-valued Homotopy algorithm. Simulation results show that the proposed algorithm converges at a very fast speed (only 4-6 snapshots in the authors simulations) and provides both excellent detection performance and easy parameter settings.