Every complex polynomial (including real polynomials) has at least one complex root. Actually, each complex polynomial of degree n has n roots, but some of those roots may be the same.

A more general concept is algebraic closure. A field F is algebraically closed if every nonconstant polynomial over F splits over F. The FTA says that C is algebraically closed. For more information on polynomials, splitting fields, and algebraic closure, consult your favourite algebra book's chapter on field extensions. For a proof of the FTA, consult any book on complex analysis---the proof is not very complicated, but it relies on more than just algebra.