Why does there exist an imaginary axis on the Argand diagram? Complex numbers can puzzle me a bit, and I think I have some gaps in my understanding that makes it confusing for me to wrap my head around. 
The way I try to explain complex numbers to myself is this:


*

*If we do not include complex numbers, we can find numbers that cannot exist on the Cartesian plane, such as with complex solutions to certain quadratics. We're missing numbers on our Cartesian plane, and we're trying to represent all numbers on this plane.

*If we instead express all numbers as $a + bi$ instead of $a$ we can now identify all numbers and be able to express them on the Cartesian plane. 


If my interpretation is wrong in some way please do explain why.
Regardless, my main question is why, this follows, that we now have the Argand diagram derivative of the reals-only Cartesian plane, with the imaginary axis on the $y$ axis and reals on the $x$ axis? Does this have to do with a logical progression of when I said "If we instead express all numbers as $a + bi$ instead of $a$ we can now identify all numbers and be able to express them on the Cartesian plane."? Let me elaborate:


*

*Since numbers are now expressed as in the form $a + bi$, if the imaginary axis is on the $y$ axis and the real number line is on the $x$ axis, all numbers can be accounted for elegantly.

*This is due to the fact that now, any number can be expressed by its projection in the imaginary and real axis. 


But due to someone establishing this rule, we now have the following properties:


*

*We now use vectors to typify numbers. $3 + 2i$ is now a projection in this plane, and we now use things like vector addition. And then that can snowball into Euler's identity?!

*Multiplying by $i$ now causes projections to "rotate" in the plane.


This just seems bizarre to me, and it's because we've elected to define our $y$ axis as a number multiplied by the square root of $-1$, which sounds totally arbitrary. 
Why couldn't I make my own diagram and instead of multiplying numbers in the y-axis by $i$, I multiply it by $k$ which I define as equal to $13$? And all these new properties I listed, like the extended Cartesian plane now being sort of like a vector space and have rotational properties seem crazy to me. 
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$$ \bbox[2px,border:2px solid red]
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So,\ to\ recap,\ my\ main\ question\ is:\ why\ has\ all\ this\ been\ implemented*,\ keeping\ in\ mind\ the\ new\ odd\ properties\ our\ extended\ Cartesian\ plane\ now\ has?\
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What's the deal about all of this? How and why did we go from our number line of reals and end up with all this?
I hope I've made my question stated clearly enough. Please let me know if something needs to be made more clear.
*And by "all this been implemented", I mean creating a $y$ axis, the imaginary axis, for the real number line and allowing all these odd properties to form, hence the title.
 A: In principle we could do what you suggest -- take $\mathbb R^2$ and associate every point $(x,y)$ to the number $x+13k$. Though the trouble with that particular plan is that each number now represents many different points -- for example, $(13,0)$ and $(0,1)$ and $(26,-1)$ are now all associated to the number $13$. This means that we can't use the scheme for anything where we calculate a number and that number points to exactly one point in the plane.
We could, however, do something more general. Take some field that extends $\mathbb R$, pick some element $\alpha$ in that field, and then represent $(x,y)\in\mathbb R^2$ by $x+\alpha y$.
As it went for $13$, if we pick $\alpha\in\mathbb R$, then we get something where a number doesn't represent a unique point. Suppose, however, that we steer clear of that case, and furthermore that we end up in the lucky situation that every element of the field represents some $(x,y)$ in the plane.
Something wonderful happens then -- namely, we can then prove (though not in the space left for me in this margin) that the field we're using must be isomorphic to $\mathbb C$ -- in other words the field is essentially the complex numbers, just called something different. In particular, somewhere in the plane there is an $(x,y)$ whose corresponding number behaves exactly like $i$.
So we could actually have said: Pick some complex number $\alpha$, and let $(x,y)$ correspond to $x+\alpha y$.  As long as $\alpha$ is not real, this will give us a perfectly good one-to-one correspondence between points and complex numbers.
Now, among all the possible choices of $\alpha$ it turns on that exactly when $\alpha=i$ or $\alpha=-i$ we get the additional nice property that multiplication by any fixed nonzero complex number will correspond to a transformation of the plane that takes geometric figures to similar geometric figures.
Having multiplication correspond to similarity transforms is a pretty nifty property, which is a reason to prefer the representation $x+iy$ over other possible $x+\alpha y$.
A: I think your difficulty starts here:

If we do not include complex numbers, we can find numbers that cannot exist on the Cartesian plane, such as with complex solutions to certain quadratics. We're missing numbers on our Cartesian plane, and we're trying to represent all numbers on this plane.

To begin with, "the" Cartesian plane that we all know and love contains points that are identified with pairs of numbers, 
specifically pairs of real numbers.
There is such a thing as a quadratic equation of two variables,
for example, $x^2 + 4xy - y^2  + 1 = 0,$
which we can plot in a Cartesian plane, but I bet that's not what you meant.
The usual introduction to complex numbers starts with equations such
as $x^2 + 1 = 0,$ for which each solution consists of just one number,
not a pair of numbers.
In fact, the reason for inventing complex numbers in the first place had nothing to do with the Cartesian plane; it took hundreds of years for those two mathematical ideas to be connected at all.
Cardano was already using $\sqrt{-1}$ as his "one weird trick" to solve cubic equations in 1545.
Descartes published his idea about coordinates of a plane in 1637--already almost a century later!--and Argand first published his diagrams in 1806.
Look at those dates. For over $250$ years after complex numbers first occurred in mathematics, nobody knew how to use a coordinate plane to represent them.
Moreover, complex numbers were not invented to solve quadratics.
Nobody cared that there were no solutions for $x$ in $x^2 + 1 = 0.$
At first, mathematicians came to grudgingly accept an occasional complex number in mathematics because they were able to make use of them while working on the problem of finding real solutions to cubic and quartic equations.
The full acceptance of complex numbers came much later.
In any case, Argand diagrams did not arise because Argand (or anyone else) was sitting around trying to think of odd properties that could be added to a Cartesian plane and dreamed this up.
They arose because the geometric interpretations of complex operations on an Argand diagram happen to exactly match what complex numbers were already known to do; those properties had all been proved by algebra with no geometric reasoning whatsoever.
