How are complex numbers, being imaginary (non real), able to represent reality (physics)? Complex numbers came from solving $x^2 = -a$, taking $i\equiv\sqrt{-1}$. It isn't representing anything real. Then how are we able to represent lots of physics using them? How can something non-existing (not real/imaginary) represent reality?
 A: It would be incorrect to class complex numbers as “non-existing”. They are just as “real” as all other numbers, and the term imaginary is slightly misleading (generally speaking).
Complex numbers are powerful mathematical tools used for solving physical problems. That is how they are able to represent physical reality. In this sense they are no more different to real numbers, functions or even more abstract mathematical objects like quarternions.
For example, consider the wavefunction in quantum physics. The wavefunction is described by an amplitude and a phase, and complex numbers encode the mathematical relationship between the amplitude and phase. Of course you could try to represent this relationship without using complex numbers, but such an endeavor would be far more complicated and involved than using complex numbers.
The use of complex numbers is similar to any other mathematical abstraction used to describe physical processes. For example, by using isomorphisms we  can map between these abstractions and their corresponding objects in physical reality.
A: Maybe this will help; when you first heard of negative numbers in your life, perhaps as a child, you might have thought that those don't represent anything real, but later you (1) found them useful as a part of a larger mathematical structure, and (2) found out that they could be mapped to something real (e.g. "take -2 steps" can be interpreted to mean "go 2 steps backwards").
Similarly, you can find things in nature that are nicely described by the properties of complex numbers. For example, complex numbers can be used to describe 2D rotations, or objects revolving around something in a plane; a real thing that happens in the real world.
The names of the components of a complex numbers are just labels; if you change "real" and "imaginary" to "foo" and "bar", it makes no difference whatsoever. These names do not imply anything about the fundamental reality of these mathematical structures.
Both the "reals" and complex numbers are no more real or imaginary than any other mathematical structure. It comes down to whether or not we can find cases where their mathematical properties are useful in describing natural phenomena (either new, or old but seen in a new light). It's just that the "real" numbers are generally more intuitive to our brains.
Some answers have said that "any and all physics can be described with real numbers alone", but that's being a bit imprecise with words. Yes, the reals can appear as fundamental objects to be manipulated in all these descriptions, but strictly speaking, the reals are just a number system, and you need a whole bunch of other mathematical structures on top of that to describe the world - vectors, tensors, fields, manifolds, symmetry groups, Hilbert spaces... So physics already uses all kinds of weird mathematical structures. That physicists sometimes make use of complex numbers either as just a mathematical gimmick to make some calculation simpler or as a way to describe something real shouldn't be that big of a deal.
Also, saying that "the result of a measurement is always real [meaning a real number]" doesn't really mean anything. The result of a measurement is some kind of a mathematical object that relates the state of the measured property of the physical object to some predefined measurement etalon1. The fact that we can generally express that as a real number, or that using real numbers is typically the most convenient option for us, is accidental, and has to do with how we as humans like to think about things, and with what's easier for us to comprehend, and with what kinds of things we can plug into our existing theoretical frameworks, computers, etc. In other words, for all we know, the choice to express measurements in terms of real numbers is arbitrary in the grand scheme of things.

1 E.g. if you measure your desk with your keyboard, and it's 2 keyboards long, you're saying that the relationship between the measured property (the length of the table) and the etalon (the keyboard) is that the table is twice as long as the the keyboard. If something is 2 m long, it's twice as long as the predefined unit of length we call the meter.
A: Imaginary is definitely a misnomer. A complex number is no more and no less real than a negative number, an irrational number and even a natural number.
Is the length of a measuring stick a real number? Does it really have an infinite number of decimal digits? Of course not. Before getting to the $\mu m$ digits you start to see that the end of the stick is not straight, but has bumps and dents, so the length of the stick is not well defined. And if you insist on leveling it with a super precise machine, at some point you get to atoms, and measuring lengths becomes real fuzzy.
If you have two apples, are they really two? Why is an apple different than the air around it? Saying that there is an apple we mark an arbitrary distinction between the atoms in the apple and the atoms in the air around it. And when there are two apples we make another arbitrary choice: we group the two single objects that we have identified into the same category. They are both apples, even though they are utterly different from each other. One has red skin, the other is yellow, one has a small worm tunnel, the other is intact...
Negative numbers are even more abstract. They are used for example when measuring temperature, but only because we have decided to fix a zero point somewhere arbitrary. In the Kelvin scale there is no use for negative numbers.
Complex numbers are as real as the other numbers. They are fantastic at describing rotations and waves, way better than real numbers.
Numbers are just a tool we use to describe the world.
A: Others already tackled the (more fundamental and crucial) topic of a misleading name. I want to stress that they are used for convenience. Any and all physics can be described with real numbers alone, but pairing two real numbers as $(a, b)$, using notation $a+bi$ (or similar) and defining some operations (using the handy $i$, which btw technically should be defined $i^2=-1$ and not with the square root; there can be issues with complex branches) and visual interpretations (arrows with real and imaginary axes) is a handy way to have more information coded in the values.
All measurements are real. But complex numbers are a useful abstraction very much like we could (and historically have) fight over how negative numbers can't signify anything real, but are of course useful and common in everyday language nowadays. Complex number are useful but not common in everyday language due to past choices in school curricula. For some reason, physics education has stuck with teaching complex numbers as a norm, the de facto mathematical expression without explaining alternatives. This choice is arbitrary and a tradition.
For comparison, take rational numbers with fraction notation. I could say a bar is $\frac{4}{3}$ meters long, and we probably wouldn't have any issues with that notion. It's clear, well-defined and we understand it. But for someone who hasn't understood ratios and fractions that notation seems to be unnecessarily complicated: Why are there two numbers, 4 and 3? Why them, what's so special about them? And they'd be asking the right questions, because we indeed shouldn't need two numbers to explain one length. (And it's not even unique, $8/6$ has the exact same value!) But it's a handy notation compared to $1,3333...$ (although both have their benefits).
A: As others have noted, complex numbers aren't especially fake in a way that would make them unphysical. But where complex numbers are physically relevant, why is this so?
Ever since classical physics, we've expected second-order ODEs that admit useful linear approximations close to a stable equilibrium to describe many physical systems. This amounts to $\ddot{x}=-\omega^2x$, with solutions linear combinations of $e^{\pm i\omega t}$. (Phase space coordinates may well be real; but with $y:=q+ip$ (non-dimensionalize everything first!), Hamilton's equations become $\dot{y}^\ast=i\frac{\partial H}{\partial y}$.) This generalized to e.g. $e^{\pm i(\omega t-kx)}$ in classical field theories' PDEs, electromagnetic plane waves being a famous consequence.
A: Complex numbers are nothing but real 2D vectors with a further operation called product with a nice interplay with the vector space structure ( they form one of the tree possible finite-dimensional real associative division algebras).
Imaginary numbers are here nothing but the vectors of the form $(0,x)$. There are some cases in physics where this structure plays some role. Not only in QM, but also in hydrodynamics or electromagnetism for instance. I think that’s all… Maybe the the choice of the name “imaginary” was unfortunate, since this mathematical structure is as real (or imaginary !) exactly as any other structure used in physics.
A: The result of a measurement is always real. Complex numbers come into play when describing states in QM. We also use complex numbers in electrodynamics, where it is just for convenience (using e-functions is easier than sin or cos), and we take then the real part of the result. In quantum mechanics, complex numbers are indeed necessary, which is in contrast to e.g. electrodynamics. However, results of measurements are still real.
