What is a simple, physical situation where complex numbers emerge naturally? I'm trying to teach middle schoolers about the emergence of complex numbers and I want to motivate this organically. By this, I mean some sort of real world problem that people were trying to solve that led them to realize that we needed to extend the real numbers to the complex. 

For instance, the Greeks were forced to recognize irrational numbers not for pure mathematical reasons, but because the length of the diagonal of a square with unit length really is irrational, and this is the kind of geometrical situation they were already dealing with. What similar situation would lead to complex numbers in terms that kids could appreciate?

I could just say, try to solve the equation $x^2 + 1 = 0$, but that's not something from the physical world. I could also give an abstract sort of answer, like that $\sqrt{-1}$ is just an object that we define to have certain properties that turn out to be consistent and important, but I think that won't be entirely satisfying to kids either.
I don't know  


  a simple, physical situation where complex numbers emerge naturally


but I can suggest a way to help you


  teach middle schoolers about the emergence of complex numbers and I
  want to motivate this organically.


I did this once as a guest lecturer in a middle school classroom by developing a geometric  interpretation of arithmetic on the number line. 

Adding a fixed number $r$ is a shift by $r$, to the right if $r &gt; 0$, to the left if $r &lt; 0$. Successive shifts add the shift amounts.  Each geometric shift is characterized by the position that $0$ moves to. You illustrate this visually by physically shifting a yardstick along a number line drawn on the board.

The answer to the question "what do you shift by so that doing it twice shifts by $r$?" is clearly $r/2$. 

This is looking ahead to square roots, but you don't say that yet.
The underlying idea is that the group of shifts is the additive group of the real numbers, but you don't say that ever.

Now that addition is done you go on to multiplication. Multiplying by a fixed positive $r$ rescales the number line. If $r&gt;1$ things stretch, if $r &lt; 1$ they shrink and multiplying by $r=1$ changes nothing.  To know what a scaling does all you need to know is the image of $1$.

Successive scalings multiply, just as successive shifts add. 
What should you do twice to scale by $9$? Half of $9$ doesn't work, but $3$ does. The class will quickly grasp that the geometric way to halve a scaling is to find the square root.

What about multiplication by a negative number? The geometry is clear: it's  reflection over $0$ followed by a scaling by the absolute value. Again the transformation is characterized by the image of $1$.

Now you're ready for the denoument. What geometric transformation can you do twice to move $1$ to $-1$ on the number line? Take your yardstick, place it on the line on the board, rotate by a quarter of a circle so that it's vertical, then another quarter and you're there. The image of $1$ is not on the line. It's at position $(0,1)$ in the cartesian coordinate system middle schoolers know about. They will find it cool to think of that point as a new number such that multiplying by it twice turns $r$ into $-r$. Name that number "$i$".

If you have brought the class along this far the rest is easy. They will quickly see the $y$ axis as the real multiples of $i$. Clearly adding $i$ should be a vertical translation by one unit. Vector addition for complex numbers follows quickly. Ask for the square root of $i$ and they will rotate the yardstick $45$ degrees. If they know about isosceles right triangles they will know that the (actually a) square root of $i$ is $(\sqrt{2}/2)(1+i)$, which they can check formally with the distributive law (which they will not ask you to prove).



A caveat. I think this should be pure fun for the class. Make that clear, so if some don't follow they don't worry. I would not try to integrate it into whatever the standard curriculum calls for. It should probably not extend over multiple class periods. Save it for a day near the end of the school year.
Not 100% sure if it is a good example, but afaik the time-dependent Schrödinger equation contains i (https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#Time-dependent_equation).