Traditional speech enhancement techniques modify the magnitude of a speech in time-frequency domain, and use the phase of a noisy speech to resynthesize a time domain speech. This work proposes a complex-valued Gaussian process latent variable model (CGPLVM) to enhance directly the complex-valued noisy spectrum, modifying not only the magnitude but also the phase. The main idea that underlies the developed method is the modeling of short-time Fourier transform (STFT) coefficients across the time frames of a speech as a proper complex Gaussian process (GP) with noise added. The proposed method is based on projecting the spectrum into a low-dimensional subspace. Experiments were carried out on the CHTTL database, which contains the digits zero to nine in Mandarin. Several standard measures are used to demonstrate that the proposed method outperforms baselines with various types of noise and SNR levels.