Imaginary numbers are just a logical extension of the real numbers a sort of mirror image of them. Complex numbers come about when you add reals to imaginary numbers.
Example of uses :
1) Solving polynomial equations. Every polynomial equation of degree n with integral coefficients will have exactly n solutions (some of which may be repeated roots). Those n roots can be all real, all imaginary/complex or a combination of real and complex roots. The complex roots will always come in conjugate pairs meaning : if a + bi is a root, then a - bi will be as well. Related to solution of polynomial equations, we have applications like finding the nth roots of unity, which you can read about. For instance, the cube root of one can have 3 values : 1, and a complex conjugate pair.
2) Simplifying various derivations : for example in finding a general form for \sin n\theta or \cos n\theta one can use De Moivre's theorem and the binomial expansion to get the expression much more easily.
3) Extending various transcendental functions : The logarithms of natural numbers can be expressed as complex numbers, based on Euler's theorem : e^{i\pi} = -1. Trigonometry can be extended beyond the usual domains by using complex numbers. By definition, \cos ix = \cosh x and \sin ix = i\sinh x. For instance arccos (2) can be computed as a complex number (actually a pure imaginary number in this case), even though you might have learned in school that the range for cosine only extends to a maximum of magnitude one.
4) Vector mathematics : A complex number can be represented as a vector on an Argand diagram, and everything that can be done in "normal" vectors can be done with a complex number representation.
5) Matrices : As an extension of vectors above, Hamilton came up with "hypercomplex numbers" and "quaternions" which are represented by square matrices.
6) Number theory : An extension of the integers are the "Gaussian integers" which use integers in the real and imaginary parts of the number. The Riemann hypothesis (which if true would have implications for the distribution of primes) uses an extension of the zeta function with complex powers.
Those are the applications that I can think of in Math. In Physics, there are various concrete applications as well : for e.g. viewing the complex number as a "phasor" is helpful in a.c. circuit analysis and wave theory (optics, electromagnetic theory). Theoretical physics uses constructs like "imaginary time", for e.g. Einstein's relativity theory uses "Minkowski space" which has an imaginary term in the time coordinate.
