Quaternion, an extension of complex number, is the first discovered non-commutative divi-sion algebra by William Rowan Hamilton in 1843. In this talk, we present the recent progress in building up the connection between the concept of quaternionic analyticity and the physics of high-dimensional topological states. Three- and four-dimensional harmonic oscillator wavefunc-tions are organized by the SU(2) Aharanov-Casher gauge potential to yield high-dimensional Landau levels possessing the full rotational symmetries and flat energy dispersions. The lowest Landau level wavefunctions exhibit quaternionic analyticity, satisfying the Cauchy-Riemann-Fueter condition, which generalizes the two-dimensional complex analyticity to three and four dimensions. It is also the Euclidean version of the helical Dirac and the chiral Weyl equations. We speculate that quaternionic analyticity provides a guiding principle for future researches on high-dimensional interacting topological states. Other progresses including high-dimensional Landau levels of Dirac fermions, their connections to high energy physics. This research is also an important application of the mathematical subject of quaternion analysis to theoretical physics, and provides useful guidance for the experimental explorations on novel topological states of matter.