"Where" exactly are complex numbers used "in the real world"? I've always enjoyed solving problems in the complex numbers during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be used/applied and hence am curious.
So what practical applications of complex numbers exist and what are the ways in which complex  transformation helps address the problem that wasn't immediately addressable?
Way back in undergrad when I asked my professor this he mentioned that "the folks in mechanical and aerospace engineering use it a lot" but for what? (Don't other domains use it too?). I'm well aware of its use in Fourier analysis but that's the farthest I got to a 'real world application'. I'm sure that's not it.
PS: I'm not looking for the ability to make one problem easier to solve, but a bigger picture where the result of the complex analysis is used for something meaningful in the real world. A naive analogy is deciding the height of tower based on trigonometry. That's going from paper to the real world. Similarly, what is it that is analyzed in the complex world and the result is used in the real world without imaginaries clouding the problem?
The question: Interesting results easily achieved using complex numbers is nice but covers a more mathematical perspective on interim results that make solving a problem easier. It covers different ground IMHO.
 A: Complex numbers are used in electrical engineering all the time, because Fourier transforms are used in understanding oscillations that occur both in alternating current and in signals modulated by electromagnetic waves.
A: Where are complex number used in the real world: iIn almost anything involving waves. Some examples are in cameras forming images, in x-ray crystallography used to determine the structure of molecules such as proteins, in MRI and CT scanners used in hospitals, in various forms of spectroscopy used to identify molecules and in lasers used to understand and predict their behaviour.
A: I was asked this exact question by my wife last night.  She was looking for an everyday example of the use of complex numbers to explain to her 8th grade math class (whose knowledge of complex numbers consists of $i = \sqrt{-1}$ ).
My response was this:

Imagine an electronic piano.  Each key produces a different tone.  A volume control changes the amplitude (volume) of all the keys by the same amount.
  That's how real numbers affect signals. 
Now, imagine a filter.  It makes some keys sound louder and some keys sound softer, depending on their frequencies. That's complex numbers -- they allow an "extra dimension" of calculation.

(Yes, I know about phase shifts and Fourier transforms, but these are 8th graders, and for comprehensive testing, they're required to know a real world application of complex numbers, but not the details of how or why.  I don't understand this, but that's the way it is)
A: Since you mentioned "real world".
The "real world" consists of miniscule particles: protons, electrons, etc. Which are not exactly particles: quantum mechanics says each of them looks like a wave. Normal waves have some "value" or "displacement" or "magnitude" in each point of space.
Magnitude (amplitude) of waves in quantum mechanics are complex! Just imagine, the whole "real world", everything you can see or touch consists of some waves with complex amplitudes!
Complex numbers are used in real world literally EVERYWHERE.
A: The other answers nicely cover specific examples of alternating current and wave equations. Basically, wherever you encounter an oscillatory phenomenon of any type, complex numbers are a natural tool to describe them easily and efficiently.
I'd like to add a related point here. The relationship between the exponential function and trigonometric functions is transparent when you use complex numbers. Damped and oscillatory motion are two sides of the same coin: they are solutions of the same (differential) equations with slightly different parameters. Varying a parameter can switch between oscillation and damping, which is related to when solutions of a quadratic equation turn from real to complex. With this example, students can "feel" the emergence of imaginary component when something starts to resonate instead of just fading out. It makes for a nice demonstration in a classroom. It helps to convince that complex numbers are not some made-up constructs but a part of nature just as reals, and make up a much more coherent theory with nicer rules and less exceptions compared to real arithmetics.
Another more dry and technical use is in equation solving in general. For instance, solving for real roots of a real polynomial can be done through complex arithmetics (with complex intermediate results). This still begs a question, where in real life you need to solve a cubic equation (as an example) but that's another story.
A: 
PS: I'm not looking for the ability to make one problem easier to solve, but a bigger picture where the result of the complex analysis is used for something meaningful in the real world. 

Well that's about what it is. It's just that the applications for complex numbers gets simpler and sometimes more elegant using them. But in fact they are not required, you could do the same thing without them.
You've got answers about electrical engineering, in which the use of complex numbers originates from the same source as in Fourier analysis. Here it's just the relationship between exponential and trigonometrics that's the reason. I'd say that Fourier analysis becomes more elegant using complex numbers and electrical engineering becomes much easier with them, but there's nothing inherent that makes you need them.
Another example is quantum mechanics. Here we have complex valued waves, but the waves themselves is not "observable", that is they will never leave the theory and escape out into reality. They are just used in the calculations, and one could probably formulate the quantum mechanics and still avoid complex numbers, but at the expense of making the theory more complex.
A: Electrical engineering with signals, for example:
http://scipp.ucsc.edu/~johnson/phys160/ComplexNumbers.pdf
A: Two-dimensional problems involving Laplace's equation (e.g. heat flow, fluid flow, electrostatics) are often solved using complex analysis, in particular conformal mapping.
A: I am afraid your conditions on what real utility of a mathematical object means is so strict as to render all applications of mathematics ridiculous. Indeed, such a restriction soon leads us into philosophising about what is useful and what is not, what is real and what is not.
Why should the compactness with which the complex numbers unify some elementary relationships (and so effect easier methods of mathematical analysis) not be counted as a real world application, since, as others have pointed out (and you yourself probably knew) they are used in the study of such periodic phenomena as the analysis of signals -- which has applications in the systems designed by engineers to convert between such signals? For that matter why should the mechanical engineer who analyses vibrations not count it a real world application when it makes him able to predict in advance the effects of the vibration of his system on a structure?
So, I hope that you can see that the idea of what is useful, real, or applicable is very flexible indeed.
Finally, let me comment on what I think the issue usually is with people finding it hard to see the practical utility (whatever that means elsewhere) of the complex number system: the positive real numbers are a natural model of the concept of quantity or magnitude, and that was the first reason for the invention of numbers (hence the negative reals and complex numbers were seen as not real, fictitious, etc. for a long time, even as recently as the nineteenth century, towards the end of which they generally came to be accepted in most places). But gradually, our idea of number evolved to include things that are not quantities (at least not in the usual sense of the term -- the discovery of antimatter is very, very recent, compared to the timeline of mathematical history) in the original understanding of the term -- eventually with the discovery of other division algebras, which began with the quaternions of Hamilton, the concept of number became nebulous and is today best left as primitive. In summary, the utility of a mathematical object should be judged in light of its nature and what it can model appropriately (which may be far removed from the ordinary experience of the common man); this nevertheless does not detract from its utility. The following is a personal opinion, but I think mathematics, no matter how rarefied and strange, is a kind of reflection of the world. I think the world is bizarre and strange, and that we have barely begun to understand and control it.
A: Complex analysis (transformation or mapping) is also used when we launch a satellite and here on earth we have $z$-plane but in space we have $w$-plane as well. So to study various factors we use transformation.
