TITLE: Why the unreasonable applicability of complex numbers in physics/engineering?
QUESTION [10 upvotes]: After years of using complex numbers in every kind of analysis of physical and electrical engineering problems I am starting to wonder: why is this particular algebra so effective in modelling the world? Because from a purely mathematical point of view $\mathbb{C}$ is no more than the field $\mathbb{R^2}$ with the operations given by:
$$(x,y) + (a,b) = (x+a, y+b)$$
$$(x,y) * (a,b) = (x*a - y*b,y*a + x*b)$$
Is there a physical motivation for these definitions which might help explain why this particular algebra is so powerful in modelling physical systems?
PS: My friend suggests that perhaps this algebra can be derived to be the algebra that is satisfied by Fourier coefficients. And that it is some kind of implicit Fourier analysis which is actually modelling the physical systems. Can that be true? 
EDIT: This question is different from the one about demystifying complex numbers. That one asks for examples for the usefulness of complex numbers. I know they are useful. I would like to know why they are useful if that is an answerable question.

REPLY [9 votes]: I think this question can be answered without appealing to quantum mechanics (and the question is more general than quantum mechanics).  
In classical physics, quantum physics, electrical engineering, chemistry, etc. you're often describing your the system at hand with differential equations.  When solving differential equations, complex numbers pop up all over the place. As alluded to in the quote above, a straightforward solution to many differential equations often makes use of complex analysis, so there's no surprise that they are used and are useful if the tools we're using come from complex analysis.  
However, we don't have to limit ourselves to just differential equations.  Pick a random polynomial and solve for all the roots.  You're far more likely to end up with complex roots than purely real roots.  If we describe the world around us in mathematical terms and everything from simple polynomials to differential equations give complex answers more often than purely real answers, it not surprising (to me at least) that complex numbers are going to show up everywhere.
A more boring answer is that complex numbers usually show up whenever phase is important (quantum mechanics, wave mechanics, electrical circuits, etc.)  You don't strictly need to use complex numbers, but it sure makes the notation a lot more compact and intuitive.  So, we use them because they're convenient.