Why was complex numbers discovered?

They were discovered when Cardano solved the third degree equation.

In the formulas that arose to solve the third degree equation, Cardano needed to take the square root of negative numbers and add them up in a certain way. The strange thing that happened was that the formulas used these complex numbers, even if the solutions to the equation where all real. This baffled the mathematicians of the time, because how could these strange numbers turn out to be "real"?

Later this was considered totally correct, when the field of complex numbers was better undestood.