Imaginary numbers appear in algebra when we try to take square roots of negative numbers.... Geometric interpretation consists in observing that two consecutive rotations of the plane by 90 degrees around a fixed point reverse the directions of the vectors. If we think of the 180-degree rotation reversing vectors as the geometric counterpart of multiplication of numbers by -1 reversing the sign, then we are inclined to accept the 90-degree rotation (of the plane containing the line of real numbers) as the square root of -1. All this looks childlishly simple, why do mathematicians make such a fuss around it? How can one dare to compare this plain idea to profound philosophical pronouncements, such as ``Cogito ergo sum'' of Descartes? But look (as my colleague David Ruelle once suggested) from another perspective. ``Cogito ergo sum'' stayed unperturbed for more than three centuries, like a monument, a Greek statue, a magnificent piece of art, impervious to the flow of time, whilst the little speck of dust, the square root of -1, have been groving and developing for hundreds of years in the minds of mathematicians, geniuses like Cauchy, Gauss and Riemann, and turned into an evergreen intensely alive vibrant tree supporting in its branches our sacred knowledge - quantum mechanics - ruling everything we see (and do not see) in this world.
(Misha Gromov, Local and global in geometry, October 29, 1999.)
The idea of an analytic function... includes the whole wealth of functions most important to science, whether they have their origin in number theory, in the theory of differential equations or of algebraic functional equations, whether they arise in geometry or in mathematical physics; and, therefore, in the entire realm of functions, the analytic function justly holds undisputed supremacy.
(David Hilbert, Mathematical Problems, 1900)