Quaternionic quantum harmonic oscillator

In this article we obtained the harmonic oscillator solution for quaternionic quantum mechanics (H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}$$\end{document}QM) in the real Hilbert space, both in the analytic method and in the algebraic method. The quaternionic solutions have many additional possibilities if compared to complex quantum mechanics (C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}$$\end{document}QM), and thus there are many possible applications to these results in future research.