See also ComplexNumbersAreYourFriends, but let's not merge them. Both titles are potentially grammatically correct; the one refers to "complex numbers" as a single collective, the other refers to each and every member of that collective.
Just to be anal retentinve - not quite. Perhaps, ComplexNumberingIsYourFriend?? "Complex numbers are your friend" is exactly as grammatically correct as "Complex numbers is your friend".
A lot of integrals are actually easier to perform using ComplexVariables?. I recommend that everyone study ComplexVariables? sometime--especially those who have to write StatisticalSoftware? or are involved in MathematicalComputing?. Besides, there needs to be a way to define the SquareRootOfNegativeOne?.
Presumably everyone gets complex numbers in high school, but ComplexAnalysis is another matter, and that's where it gets interesting.ComplexNumbers simply simplify so many things, since they allow you to see the whole picture. From a SignalProcessing? perspective, it all has to do with HarmonicOscillation? not having a solution of order 1 over the RealNumberField?.
See also ComplexNumbers, FortranLanguage, PythonLanguageCategoryMath