A complex number is an ordered pair which obeys a special set of rules for multiplication.
The usual visualization is the complex plane: the first axis is a real number, the second axis is an imaginary number.
Then given two complex pairs (a,b) and (c,d) the multiplication rule is:
(a,b) x (c,d) = (a x c - b x d, a x d + b x c), which is what you would get if you were to write it out as
(a + bj) x (c + dj) and treat jxj=-1, and regroup the resulting set of terms as real and imaginary as a pair.
The resulting algebra is commutative and associative, but the complex numbers are not "ordered" ... you cannot say that (a,b) is greater or lesser than (c,d), though you can determine the magnitudes (distance from the origin of the plane) ... then all complex numbers lying on the same circle have the same magnitude. Two complex numbers are equal if corresponding elements of each pair are equal.