Higher-order exchange interactions and quantum effects are widely known to play an important role in describing the properties of low-dimensional magnetic compounds. Here we identify the recently discovered two-dimensional (2D) van der Waals (vdW) CrI3 as a quantum non-Heisenberg material with properties far beyond an Ising magnet as initially assumed. We find that biquadratic exchange interactions are essential to quantitatively describe the magnetism of CrI3 but requiring quantum rescaling corrections to reproduce its thermal properties. The quantum nature of the heat bath represented by discrete electron-spin and phonon-spin scattering processes induced the formation of spin fluctuations in the low temperature regime. These fluctuations induce the formation of metastable magnetic domains evolving into a single macroscopic magnetization or even a monodomain over surface areas of a few micrometers. Such domains display hybrid characteristics of Neel and Bloch types with a narrow domain wall width in the range of 3-5 nm. Similar behaviour is expected for the majority of 2D vdW magnets where higher-order exchange interactions are appreciable.