I need to calculate Bessel and Spherical Bessel functions with real orders and complex arguments. Is there a way to define complex Bessel functions in terms of the real Bessel functions in NR? (I'm a programmer, not a mathematician, so forgive me if the question doesn't even make sense.)

Your question does make sense, and the answer to it is "no." There is no expression for J_n(x+iy) in terms of x, y, and other standard functions. It's simply a new and different beast.

I believe you will find some routines at NETLIB for Bessel functions of complex arguments.