Several scientists learn only a first course in complex analysis, and hence they are not familiar with several important properties: every polygenic function defines a congruence of clocks; the basic properties of algebraic functions and abelian integrals; how mankind arrived at a rigorous definition of Riemann surfaces; the concepts of dianalytic structures and Klein surfaces; the Weierstrass elliptic functions; the automorphic functions discovered by Poincare' and their links with the theory of Fuchsian groups; the geometric structure of fractional linear transformations; Kleinian groups; the Heisenberg group and geometry of the complex ball; complex powers of elliptic operators and the theory of spectral zeta-functions; an assessment of the Poincare' and Dieudonne' definitions of the concept of asymptotic expansion. The book is unique both for the selection of topics and for the readable access that it offers to the otherwise too large landscape of modern complex analysis.