Banach's Isometric Conjecture over the Complex Field

We complete Banach’s isometric conjecture over the complex field. More precisely, if $X$ is a complex normed space and, for some $2⩽n<{\dim }_{\text{C}}X$, all its $n$-dimensional complex subspaces are isometric as metric spaces, then the norm is induced by a Hermitian inner product. We also prove the quaternionic counterpart. The central geometric argument first treats real star bodies without convexity or central symmetry; applied to circled complex or quaternionic bodies, it shows that mutually real-linearly equivalent hyperplane sections force the ambient body to be a Hermitian ellipsoid. The proof adapts the bundle-degree mechanism introduced by Lu and Yang for the real case. Finally, we obtain extensions to absolutely homogeneous functions, graded Fréchet spaces, metrisable locally convex spaces, and compatible translation-invariant metrics.