Why the notation for complex numbers? Why do we use the a + bi notation for complex numbers? We don't use that notation for any other two dimensional quantity do we? If not, why use it exclusively for complex numbers?
 A: A complex number is by definition a "number" composed by a real part and an imaginary part. 
$$z = a + ib$$
or also
$$z = x + iy$$
or whatever you want.
The notation is comfortable because it makes you to see immediately what the real and imaginary parts be.
$$a = \text{real part}$$
$$b = \text{imaginary part}$$
It's simply... effective! Immediate.
P.s. What other two dimensional quantities? 
A: It's not just notation -- $a+bi$ literally means to take the two real numbers, multiply one of them by $i$ and add the result to the other real number. These two operations are the same multiplication and addition operations that work for complex numbers in general, so once we know how to write a real number and have given a name to the constant $i$, no additional definitions are necessary to make sense of the $a+bi$ notation.
This fortunate situation is not present for many other "two-dimensional quantities", and therefore they need some ad-hoc notation for combining their two components into a single quantity.  But there is no need to go to such detours for complex numbers.

Note, however, that in particular in engineering texts it seems to be somewhat common to write 2-dimensional vectors from $\mathbb R^2$ not as $(a,b)$ but as $a\hat\imath+b\hat\jmath$ with $\hat\imath$ and $\hat\jmath$ being conventional names for basis vectors -- which is more or less the same principle as for $a+bi$.
A: A complex number is not a two dimension thing, it is a certain type of number. You can find a isomorphism between the complex numbers and the points on the Cartesian plane, i.e. every complex number has a representation on the Argand plane but that doesn't mean its two dimensional. For two dimensional things we often use the notation $(x,y)$. Here $x$ and $y$ are numbers and depending on the situation could be various types of numbers, e.g. natural numbers if we are naming streets/avenues in a city.
