  Complex determinants, Gramians and blades (simple or decomposable
multivectors) are related to volumes, but not in the same way as real ones. We
present their complete geometric interpretations, including complex
orientations. We also link real and complex concepts via a reality index, which
measures the lack of holomorphy of real subspaces of $\mathbb{C}^n$. These
results lead to volumetric Pythagorean theorems, a new complex Hodge star, and
interpretations for Grassmannian metrics, among other uses.
