The notion is entirely mathematical with no physics implied, but should be seen as a tool, for instance, as exemplified by the Wick rotation.
The spacetime underlying Albert Einstein's field equations, which mathematically describe gravitation, is a real 4 dimensional pseudo-Riemannian manifold.
Its appearance in physics can be rooted to attempts of unifying the fundamental interactions, originally gravity and electromagnetism.
[1][2] Other ideas include mapping real spacetime into a complex representation space of SU(2, 2), see twistor theory.
In 1926, Oskar Klein suggested[5] that Kaluza's extra dimension might be "curled up" into an extremely small circle, as if a circular topology is hidden within every point in space.
In 1932, Hsin P. Soh of MIT, advised by Arthur Eddington, published a theory attempting to unify gravitation and electromagnetism within a complex 4-dimensional Riemannian geometry.
In the latter years of World War II, Einstein began considering complex spacetime geometries of various kinds.
[7] In 1953, Wolfgang Pauli generalised[8] the Kaluza–Klein theory to a six-dimensional space, and (using dimensional reduction) derived the essentials of an SU(2) gauge theory (applied in quantum mechanics to the electroweak interaction), as if Klein's "curled up" circle had become the surface of an infinitesimal hypersphere.