Quaternion is an important extension of complex number,which was the first discovered non-commutative division algebra by Hamilton in 1843.In this talk,we will apply quaternion wave-functions to describe topological properties of quantum mechanical systems including Bose-Einstein condensations(BECs)with synthetic gauge fields,and the high dimensional topological insulators based on the SU(2)generalization of Landau levels.The conventional condensation wave-functions of bosons are positive-definite required by the “no-node” theorem as illustrated in Feynmans lectures of statistical mechanics.This theorem significantly reduces the possibility of finding novel physics in ground states of bosons including 4He and many ultra-cold alkali atom BECs.However,such a constraint is released in spin-orbit coupled BECs,and we will show that the corresponding condensation wave-functions are not only complex-valued but also best interpreted as quaternion-valued.They exhibits topological defects of quaternonic phase distributions as 3D skyrmions which are the SU(2)generalization of the usual Abrikosov vortex lattice.In the second part of the talk,we will generalize the usual 2D Landau levels in the external magnetic fields to 3D with the fully rotational symmetry and time-reversal symmetry.It is well-known that the complex-analyticity of the 2D lowest Landau levels is essential for the construction of the Laughlin states for the fractional quantum Hall states.We have proved that the lowest 3D Landau level states satisfy the quaternionic analyticity condition.We anticipate that this elegant analytic property and the also the energy spectra flatness of 3D Landau levels will also facilitates the study of high dimensional fractional topological states.