  Quaternion, an extension of complex number, is the first discovered
non-commutative division algebra by William Rowan Hamilton in 1843. In this
article, we review the recent progress on building up the connection between
the mathematical concept of quaternoinic analyticity and the physics of
high-dimensional topological states. Three- and four-dimensional harmonic
oscillator wavefunctions are reorganized 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 {\it
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. After dimensional reductions,
these states become two- and three-dimensional topological states maintaining
time-reversal symmetry but exhibiting broken parity. We speculate that
quaternionic analyticity can provide 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, and high-dimensional Landau levels in the Landau-type gauges,
are also reviewed. This research is also an important application of the
mathematical subject of quaternion analysis in theoretical physics, and
provides useful guidance for the experimental explorations on novel topological
states of matter.
