The modeling of human emotion expression in speech signals is an important, yet challenging task. The high resource demand of speech emotion recognition models, combined with the the general scarcity of emotion-labelled data are obstacles to the development and application of effective solutions in this field. In this paper, we present an approach to jointly circumvent these difficulties. Our method, named RH-emo, is a novel semi-supervised architecture aimed at extracting quaternion embeddings from real-valued monoaural spectrograms, enabling the use of quaternion-valued networks for speech emotion recognition tasks. RH-emo is a hybrid real/quaternion autoencoder network that consists of a real-valued encoder in parallel to a real-valued emotion classifier and a quaternion-valued decoder. On the one hand, the classifier permits to optimize each latent axis of the embeddings for the classification of a specific emotion-related characteristic: valence, arousal, dominance and overall emotion. On the other hand, the quaternion reconstruction enables the latent dimension to develop intra-channel correlations that are required for an effective representation as a quaternion entity. We test our approach on speech emotion recognition tasks using four popular datasets: Iemocap, Ravdess, EmoDb and Tess, comparing the performance of three well-established real-valued CNN architectures (AlexNet, ResNet-50, VGG) and their quaternion-valued equivalent fed with the embeddings created with RH-emo. We obtain a consistent improvement in the test accuracy for all datasets, while drastically reducing the resources' demand of models. Moreover, we performed additional experiments and ablation studies that confirm the effectiveness of our approach. The RH-emo repository is available at: https://github.com/ispamm/rhemo.