  Let $\mathfrak{g}$ be a complex semisimple Lie algebra. We give a description
of characters of irreducible Whittaker modules for $\mathfrak{g}$ with any
infinitesimal character, along with a Kazhdan-Lusztig algorithm for computing
them. This generalizes Milicic-Soergel's and Romanov's results for integral
infinitesimal characters. As a special case, we recover the non-integral
Kazhdan-Lusztig conjecture for Verma modules.
