  Complex numbers enter fundamental physics in at least two rather distinct
ways. They are needed in quantum theories to make linear differential operators
into Hermitian observables. Complex structures appear also, through Hodge
duality, in vector and spinor spaces associated with space-time. This paper
reviews some of these notions. Charge conjugation in multidimensional
geometries and the appearance of Cauchy-Riemann structures on Lorentz manifolds
with a congruence of null geodesics without shear are presented in considerable
detail.
