Recurrence techniques are described for the generation of Bessel functions of the first and second kinds where both the argument and order may be complex. The method is shown to be accurate for several well known forms of functions, including Kelvin and spherical Bessel functions. The accuracy of the general case of complex order and argument is determined by computing the Wronskians and by verifying some addition theorems of the Bessel functions over wide ranges of order and argument. Procedures for the accurate generation of complex-argument gamma functions are also described.