A novel quaternion-valued singular spectrum analysis (SSA) is introduced for multichannel analysis of electroencephalogram (EEG). The analysis of EEG typically requires the decomposition of data channels into meaningful components despite the notoriously noisy nature of EEG--which is the aim of SSA. However, the singular value decomposition involved in SSA implies the strict orthogonality of the decomposed components, which may not reflect accurately the sources which exhibit similar neural activities. To allow for the modelling of such co-channel coupling, the quaternion domain is considered for the first time to formulate the SSA using the augmented statistics. As an application, we demonstrate how the augmented quaternion-valued SSA (AQSSA) can be used to extract the sources, even at a signal-to-noise ratio as low as -10 dB. To illustrate the usefulness of our quaternion-valued SSA in a rehabilitation setting, we employ the proposed SSA for sleep analysis to extract statistical descriptors for five-stage classification (Awake, N1, N2, N3 and REM). The level of agreement using these descriptors was 74% as quantified by the Cohen's kappa.