We introduce HyperGaussians, a novel extension of 3D Gaussian Splatting for high-quality animatable face avatars. Creating such detailed face avatars from videos is a challenging problem and has numerous applications in augmented and virtual reality. While tremendous successes have been achieved for static faces, animatable avatars from monocular videos still fall in the uncanny valley. The de facto standard, 3D Gaussian Splatting (3DGS), represents a face through a collection of 3D Gaussian primitives. 3DGS excels at rendering static faces, but the state-of-the-art still struggles with nonlinear deformations, complex lighting effects, and fine details. While most related works focus on predicting better Gaussian parameters from expression codes, we rethink the 3D Gaussian representation itself and how to make it more expressive. Our insights lead to a novel extension of 3D Gaussians to high-dimensional multivariate Gaussians, dubbed 'HyperGaussians'. The higher dimensionality increases expressivity through conditioning on a learnable local embedding. However, splatting HyperGaussians is computationally expensive because it requires inverting a high-dimensional covariance matrix. We solve this by reparameterizing the covariance matrix, dubbed the 'inverse covariance trick'. This trick boosts the efficiency so that HyperGaussians can be seamlessly integrated into existing models. To demonstrate this, we plug in HyperGaussians into the state-of-the-art in fast monocular face avatars: FlashAvatar. Our evaluation on 19 subjects from 4 face datasets shows that HyperGaussians outperform 3DGS numerically and visually, particularly for high-frequency details like eyeglass frames, teeth, complex facial movements, and specular reflections.