The recently introduced multiscale entropy (MSE) method accounts for long range correlations over multiple time scales and can therefore reveal the complexity of biological signals. The existing MSE algorithm deals with scalar time series whereas multivariate time series are common in experimental and biological systems. To that cause, in this paper the MSE method is extended to the multivariate case. This allows us to gain a greater insight into the complexity of the underlying signal generating system, producing multifaceted and more robust estimates than standard single channel MSE. Simulations on both synthetic data and brain consciousness analysis support the approach.