We investigate the non-Gaussian features of the intergalactic medium (IGM) at redshift z similar to 5-6 using the Ly alpha transmitted flux of quasar absorption spectra and a cosmological hydrodynamic simulation of the concordance Lambda CDM universe. We show that the neutral hydrogen mass density field and Ly alpha transmitted flux fluctuations possess all the non-Gaussian features predicted by the log-Poisson hierarchy. This depends only on two dimensionless parameters beta and gamma, describing, respectively, the intermittence and singularity of the random fields. We find that the non-Gaussianity of the Ly alpha transmitted flux of quasars from z = 4.9 to z = 6.3 can be well reconstructed by the hydrodynamical simulation samples. Although the Gunn-Peterson optical depth and its variance undergoes a significant evolution in the redshift range of 5-6, the intermittency measured by beta is almost redshift-independent in this range. More interestingly, the intermittency of the quasar's absorption spectra on physical scales 0.1-1 h(-1) Mpc in redshift 5-6 is found to be about the same as that on physical scales 1-10 h(-1) Mpc at redshifts 2-4. Considering the Jeans length is less than 0.1 h(-1) Mpc at z similar to 5, and 1 h(-1) Mpc at z similar to 2, these results imply that the non-linear evolution in high and low redshifts will lead the cosmic baryon fluid to a state similar to fully developed turbulence. The log-Poisson high-order behaviour of the current high-redshift data of a quasar's spectrum can be explained by the uniform ultraviolet background in the redshift range considered. We have also studied the log-Poisson non-Gaussianity by considering an inhomogeneous background. With several simplified models of the inhomogeneous background, we have found that the effect of the inhomogeneous background on the log-Poisson non-Gaussianity is no larger than 1 sigma.