In the expression $a+it$ for a complex number, is the "$+$" just there to join the real and imaginary parts? or do the rules of arithmetic apply? We know a complex number is given as:
$$a + it \quad\text{where}\quad i= \sqrt {-1} $$
"$a$" and "$t$" are both real numbers, and "$i$" is an imaginary unit. (Complex numbers can be defined as ordered pairs $(x, y)$ of real numbers that are to be interpreted as points in the complex plane.)

What does the addition sign ("$+$") have to do with this expression? 

Was the addition sign put just to keep these so-called "real" and "imaginary" parts together, as in vectors? Are the rules of arithmetics really applicable here? For example,


*

*multiplication of two complex numbers
$$(2 + 2i)(4 + 2i) $$

*can we say that
$$ n^{a+it} = (n^a)(n^{it}) $$ 

The original reason for the $x + iy$ notation probably dates from a day of less formal and more intuitive notation.  
A more contemporary reason is to define the complex numbers formally as ordered pairs $(x, y)$ but then observe that the subset $(x, 0)$ is isomorphic to the real numbers.  We now redefine the reals to be this subset and use just $x$ as a shorthand for $(x,0)$.  Next we define $i$ to be $(0,1)$.  We then find that $(x,y) = x(1,0) + y(0,1) = x +iy$.  So, $x + iy$ is just a convenient and familiar way to write $(x,y)$.
Similar steps happen when the integers are defined from the natural numbers, the rational numbers from the integers, and the reals from the rationals.  Strictly speaking the complex number $1$ is not the same as the real number $1$ and that is not the same as the rational $1$, the integer $1$, or the natural number $1$.  Each time we extend, we redefine the old smaller system as a subset of the new isomorphic to it.  Most of the time this is not confusing and makes life and notation simpler.  Occasionally, you need to remember this.  
