What is the mathematical nature of $i$? It is well known that $i$ is unit imaginary part of any complex number, but many uses of $i$ show that has others mathematical properties, for example in integration area, if I want to compute integral of $ix$ I will get $i \frac{x²}{2} $ then here $i$ is considered a constant. Also if I want to check divisors of $i$ I got only  $1$ then here $i$ has divisors however it is not integer.

Really I would like to know more about the nature of "unit imaginary part" $i$, or what the Mathematical nature of $i$ is?  

Thank you for any help.
$i$ is an abbreviated form for $(0,1)$.

Its introduction is particularly useful because it lets you remember what the rules of complex product are.

E.g. $i^2=-1$ is more simple to remember than $(0,1) \dot (0,1) = (-1, 0)$
Also $(a+ib)(c+id)=ac-bd+i(ad+bc)$ is more simple to remember than $(a,b)\dot (c,d) = (ac-bd, ad+bc)$
$i$ is an abbreviated form for $(0,1)$.

Its introduction is particularly useful because it lets you remember what the rules of complex product are.

E.g. $i^2=-1$ is more simple to remember than $(0,1) \dot (0,1) = (-1, 0)$
Also $(a+ib)(c+id)=ac-bd+i(ad+bc)$ is more simple to remember than $(a,b)\dot (c,d) = (ac-bd, ad+bc)$