>>12296411C is one of the greatest objects in mathematics. There's nothing mystical about it, the complex numbers are as real as the real numbers.
There are many ways to actually construct C from the reals R. One is as follows:
Let C be the set of pairs (a,b) with a,b real numbers and with operations
(a,b)+(c,d)=(a+b, c+d)
(a,b)*(c,d)=(ac-bd, bc+ad).
We usually write a+bi for (a,b) or just a when b=0 or just bi when a=0.
This definition lets you do arithmetic on the complex numbers. It turns out that i*i=-1.
What's special about C is that actually it's a field that contains R. It means that any two elements of C can be divided (if the second one is nonzero) and that the operations + and * satisfy all the usual axioms of arithmetic such as
ab=ba
a(b+c)=ab + ac
etc.
The reason why C is so cool and the reason why it pops up literally everywhere is because of the fundamental theorem of algebra (misnomer, it's actually a theorem in analysis), which says that any nonconstant polynomial p(x) with coefficients in C (possibly R) has a root in C, that means there is a complex number z such that p(z)=0.
From this it follows that you can write any polynomial as a product of linear factors.
In particular, whenever you have a real polynomial you can explain its behavior very well by extending the domain to the complex numbers.
Aside from this fundamental theorem, another reason for considering the complex numbers is that most of the elementary functions like cos, sin, tan, log, e^x, sqrt, and so on are actually much better understood when viewed as functions on the complex numbers. Their analytic properties and integrals become much more tractable. You often see real improper integrals of elementary functions being calculated with complex analysis by complex integration.
