What is a complex number?

A complex number is one that is made up of a complex and real part, they can all be written as:
a+bi, where a and b are real numbers.
The thing that makes the number "complex" is "i", the imaginary number. "i" is the square root of -1, so if you had "square root of -4" your answer would be "root 4 times by i" or "2i".
It has many applications, particularly in Pure Mathematics.