Painleve - It came to appear that, between two truths of the real domain, the easiest path quite often passes through the complex domain.
Hadamard - It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one."
Read first "The Magic Wand Theorem of A. Eskin and M. Mirzakhani" by Anton Zorich for motivation on Riemann sufaces.
And, "A Singular Mathematical Promenade" by Etienne Ghys (particularly p. 87-93 and glance at the Wikipedia article on monodromy).
More prosaically:
To understand integral transform solutions (Mellin, Laplace, Fourier) to pdes, you need to understand poles and branch cuts of complex functions.
Solutions to Laplace's equation in two dimensions are called harmonic functions that are the real and complex components of a complex function and give mutually orthogonal contour lines on the complex plane and Riemann sphere.
From "THE THEOREM OF RIEMANN-ROCH AND ABEL’s THEOREM" by Siu:
The theorem of Riemann-Roch and Abel’s theorem could be interpreted as answering the question: for which configuration of charges, dipoles, or multipoles on a compact Riemann surface of genus ≥ 1 would the flux functions (whose level curves are the flux lines and which are the harmonic conjugates of the electrostatic potential functions) in the case of the theorem of Riemann-Roch, or their exponentiation after multiplication by 2πi in the case of Abel’s theorem, be single-valued on the Riemann surface so that the flux lines are closed curves?
At a more basic level for numerical analysis, in understanding convergence of real power series and, therefore, series solns. to pdes, you need to understand singularities (poles and branch cuts in the complex domain) and these involve Riemann surfaces.
Same for Newton (finite difference) and sinc function (Nyquist-Shannon) interpolations of sequences of real/complex numbers and their numerical analytic continuations and for asymptotic series a la Poincare. (Norlund, Poincare, and Berry wrote well on these topics.)
Helps in understanding convolutions, Dirac Delta functions and their derivatives, and, therefore, fractional calculus and operational calculus.
Necessary in understanding basic string theories.
The list is endless. Without such knowledge, you live (perhaps blissfully) in Abbott's Flatland.
The examples really suggest that you may be imposing a gratuitous, restrictive dichotomy--there is plenty of synergy between the study of numerical analysis and Riemann surfaces and both provide paths to other intriguing areas of the grand, evolving tapestry of mathematics, engineering, and science. (Of course, if you are looking where the money is in America, well I suggest a medical degree or starting a munitions factory.)
