What does imaginary number maps to physically? I am taking undergraduate quantum mechanics currently, and the concept of an imaginary number had always troubled me. I always feel that complex numbers are more of a mathematical convenience, but apparently this is not true, it has occurred in way too many of my classes, Circuits, Control Theory and now Quantum Mechanics, and it seems that I always understand the math, but fail to grasp the concept in terms of its physical mapping. Hence my question, what does imaginary number maps to physically?
Any help would be much appreciated
 A: Phase and amplitude! While complex numbers are used in the analysis of Classical Harmonic Oscillator or the LCR circuit as a part of Mathematical technique, wherein once the solution is obtained the real part is extracted. However in Quantum mechanics they play a central role, and QM cannot be formulated without them.
We necessarily need both the concept of Phase and Amplitude in the description of quantum mechanics. When 2 event are separated by multiple paths, then the information about later event occurring given first happens is encoded in the Probability Amplitude. Amplitude of a Single path is essentially the quantity $e^{iS}$, where S is the action. To calculate the Probability amplitude sum over the amplitude for each and every possible path in which the final event can be realized if the Initial event had happened. The Probability that the final event happens is Modulus square of the probability amplitude.
The use of Complex numbers(And Probability Amplitudes) in Quantum Mechanics follows very naturally as a generalization of the concept of Probability, And Feynman very nicely Illustrates this in his paper, "Space-Time Approach to Non-Relativistic Quantum Mechanics" or "The Concept of Probability in Quantum Mechanics"
While one can always rewrite all the formulas in Quantum mechanics as 2 real numbers, However then the beauty, and its simplicity is lost.
A: As far as describing nature is concerned mathematics is another language that can describe it.
In ancient times there was no mathematics other than arithmetic, and that was the way nature was described: cycles of the moon, trajectories of planets etc.
Then came geometry and it used mathematical tools to describe nature further, based on arithmetic but using geometrical axioms and theorems. 
Algebra came late, from the arabs, during the middle ages where the dogmatism of the church had stopped progress in science. With enlightment, mathematics took off and became a beautiful tool describing/modeling nature by the time of Newton, the co-inventor of calculus. Negative numbers and complex numbers became necessary by using algebra and are now incorporated in all mathematics useful for describing nature, as you have found out.
We can plot negative numbers in an x plot, if that is what you mean by "maps to physically". Complex  numbers, one axis real one imaginary are another plot where the solutions for physical systems can be displayed economically. They simplify notations and calculations. A relevance to physics problems comes in that they describe/incorporate trigonometric functions which continually appear in solutions of physics boundary problems due to the harmonic nature of many set ups. They are economical, in the same way that algebra became economical, not needing the convoluted sentences of the classical (BC) times mathematics.  
Complex numbers for physics are, as all mathematical tools used, a convenient representation fitting the data. 
A: It's just a mathematical tool1, which is especially useful to portray some kinds of vectors and sine waves. It doesn't have a real "meaning" here, and we can do fine whthout it (calculations just become more tedious)
For example, in circuit analysis, we convert the coordinate pair $(V,\phi)$ (which isn't linear; adding two voltages need not add their phases) into the coordinate pair $(V\cos\phi,V\sin\phi)$, which is linear. Now, instead of writing it as a coordinate pair, we write it as a complex number (which is essentially just a pair of numbers) and then notice that we can extend this to the concept of impedance and even be able to divide by these numbers and get the correct answer. Writing voltage as $(V\cos\phi)\hat i +(V\sin\phi)\hat j$ would still give a linear system, but we can't divide by vectors so it's limited in application.
In quantum mechanics, the use is similar -- we are expressing a linear quantity with a phase. Again, this is a nonlinear coordinate pair, but rewriting it as a complex number gives us a linear coordinate pair that plays nice with multiplication and division. When we bring linear algebra into it,  one sees that the representation is even more apt.
Again, it's just a tool or a representation. It has no real meaning.
1. As @MBN mentioned below, real numbers are a mathematical tool, too. We can't "directly" measure real numbers, either. 
A: If you really wanted to, you could formulate the laws of physics without using imaginary numbers - see, for example, Can one do the maths of physics without using $\sqrt{-1}$?
Let's say you need to do a fourier decomposition of a function $f$ in order to find how some responds to being driven at $f$.
If you think back, fourier decomposition is normally introduced with only real numbers, and a function $f$ was decomposed into odd and even parts, which can be represented as infinite series of cosines and sines respectively, and you could certainly do that. However, if we view $f$ as the real part of some complex-valued function, the equations become much simpler. 
If this makes you wonder whether the complex-valued function is "more real" than the real part, and what is the physical significance, you could see $f$ as $\frac{1}{2}(z + \bar{z})$ where $z$ is the complex function; then, $z$ and $\bar{z}$ both have no physical significance, but they formally satisfy the equations governing the system, hence their sum must also.
A: @Prathyush's answer is completely correct and provides  nice reference, but I just wanted to add that behind his discussion about phases and path integrals is the concept of unitary and conservation of probability, which is a physical concept, and the OP asked about what imaginary numbers "map to physically". Thinking in the Heisenberg picture one can write every operator as a real diagonal matrix and a set of mappings (rotations) between these operators, but in order for everything to conserve probability it is absolutely essential that these rotations contain imaginary numbers in general as any unitary operator may be written as $\exp{i \hat{H}}$, where $\hat{H}$ is a Hermitian operator (with real eigenvalues). It is similar for another very special rotation in Hilbert space, which is time evolution, without which there would be no physics as such. This has a deep connection to the relation between Lie algebras and Lie groups. 
Without complex numbers we would have no unitarity and no conservation of probability or would be led to a trivial quantum theory which is not dynamical!
A: Both sides of a complex number are each real numbers. Either the real part or the imaginary part can be used for computing the value of a measurable quantity. And the results are always the same. Then we are left with only 'i', to determine it's Mathematical and then physical meaning. The allegory of 'i' harbors a real Mathematical meaning which when unveiled, discloses the 'reality' of a 'complex' number. In other words, 'complex numbers' are 'real numbers' in disguise.
One can say the above in another way by saying that (-1)^1/2 is completely a real number.
A: A rotation by a right angle. So complex numbers come up whenever you have periodicity/oscillations/phases etc.
For a simple example, consider the physical intuition for Euler's formula -- if you push something with force $F=kx$, you get exponential motion, but $F=-kx$ is periodic motion. The latter, where you have periodic, trigonometric functions, is where there are complex numbers/imaginary exponents.
Similarly when describing light, the polarisation of a light wave behaves as a phase. The same idea carries over to quantum mechanics.
