  The paper presents an ab initio account of the paraxial complex geometrical
optics (CGO) in application to a scalar Gaussian beam propagation and
diffraction in a 3D smoothly inhomogeneous medium. The paraxial CGO deals with
quadratic expansion of the complex eikonal and reduces the wave problem to the
solution of ordinary differential equations of Riccati type. This substantially
simplifies description of Gaussian beams diffraction as compared to full wave
or parabolic (quasi-optics) equations. For a Gaussian beam propagating in a
homogeneous medium or along the symmetry axis in a lens-like medium, the CGO
equations possess analytical solutions, otherwise they can be readily solved
numerically. As a non-trivial example we consider Gaussian beam propagation and
diffraction along a helical ray in an axially symmetric waveguide medium. It is
shown that the major axis of the beam's elliptical cross-section grows
unboundedly; it is oriented predominantly in azimuthal (binormal) direction and
does not obey the parallel transport law.
