In this paper, we obtain the analytical solution of the off-waist inputted complex argument Laguerre-Gaussian beams and their mean squared beam width in nonlocal nonlinear media. The propagation of the complex argument Laguerre-Gaussian beams in the nonlocal nonlinear media is investigated in detail. The examples show that the pattern shape of a (n,m) mode complex argument Laguerre-Gaussian beam varies periodically with the period Δz=πzc in strongly nonlocal nonlinear media if n≠0.But if n=0, its pattern shape remains unvaried and the beam width varies periodically during propagation. Under the off-waist incident condition, the propagation of the (0,m)mode complex argument Laguerre-Gaussian beam behaves as a breather during propagation, no matter what the power of the incident beam is. Only when the beam is input at the waist and the input power equals the critical power would the breather be reduced to a soliton.