TITLE: Most natural intro to Complex Numbers
QUESTION [7 upvotes]: This is a soft question but I'm willing to ask.
There are few ways to introduce the field of complex numbers, but if You had the opportunity to write an elementary textbook, what would be the most intuitive introduction for a very sceptical audience?
Example of an "bad" answer: write down all the axioms and then quote Hilbert "existence in mathematics means freedom from contradiction".

REPLY [6 votes]: I've had some fun and some success with middle school kids using this approach.
They know about the number line. They can understand that addition of a fixed quantity can be thought of as translation, right for positives and left for negatives. Note that translation by $a$ takes the number $0$ to $a$. They can understand a fraction like $a/2$ as what you need to translate by twice to translate by $a$.
Then we move on to multiplication by positive numbers. As a transformation, multiplying by a fixed $a$ is a change of scale. Note that scaling by $a$ takes the number $1$ to $a$. That leads to a definition of $\sqrt{a}$ as the number you have to scale by twice to scale by $a$.
Next we note that multiplication by $-1$ is a reflection about $0$ on the line. (Incidentally, that helps with "why  is a negative times a negative a positive?") But what is half that reflection? What can you do twice to end up moving $x$ to $-x$? Just rotate the number line half a turn (counterclockwise is the right convention). Kids really like that idea. By analogy, that's a rescaling by the number that $1$ moves to - so we have $i$ in the right place in the newly introduced number plane. The additive structure of the rest of the number plane follows easily when interpreted as translation. 
The full multiplicative structure is much harder - both historically and for now tired middle schoolers. You can do it with the distributive law, but the geometry is harder. You can't quite get to polar coordinates. But they will be delighted to find $\sqrt{i}$ as a rotation and then its real and imaginary parts since they know the length of the hypotenuse of an isosceles right triangle with side $1$ is $\sqrt{2}$.