A matrix is called a complex-L matrix if its complex sign pattern implies that it is of full column rank.The definition is a generalization of L-matrices from real field to complex field.The recognition problem of complex-L matrices is studied in this paper, a combinatorial characterization of complex-L matrices is given, showing that the problem of recognizing complex-L matrices is in the class co-NP-complete.It is also shown this recognition problem can be reduced to its subproblem for which each entry of the matrices is either real or pure imaginary.