Quaternion　algebra　is　a　noncommutative　associative　algebra.　Noncommutativity　limits　the　flexibility　of　computation　and　makes　analysis　related　to　quaternions　nontrivial　and　challenging.　Due　to　its　applications　in　signal　analysis　and　image　processing,　quaternionic　Fourier　analysis　has　received　increasing　attention　in　recent　years.　This　paper　addresses　phase　retrievability　in　quaternion　Euclidean　spaces　HM\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$${\mathbb　{H}}＜^＞{M}$$\end{document}.　We　obtain　a　sufficient　condition　on　phase　retrieval　frames　for　quaternionic　left　Hilbert　module　(HM,(center　dot,center　dot))\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\big　({\mathbb　{H}}＜^＞{M},\,(\cdot　,\,\cdot　)\big　)$$\end{document}　of　the　form　{emTng}m,n　is　an　element　of　NM\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\{e_{m}T_{n}g\}_{m,\,n\in　{\mathbb　{N}}_{M}}$$\end{document},　where　{em}m　is　an　element　of　NM\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\{e_{m}\}_{m\in　{\mathbb　{N}}_{M}}$$\end{document}　is　an　orthonormal　basis　for　(HM,(center　dot,center　dot))\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\big　({\mathbb　{H}}＜^＞{M},\,(\cdot　,\,\cdot　)\big　)$$\end{document}　and　(center　dot,center　dot)\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$(\cdot　,\,\cdot　)$$\end{document}　is　the　Euclidean　inner　product　on　HM\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$${\mathbb　{H}}＜^＞{M}$$\end{document}.　It　is　worth　noting　that　{em}m　is　an　element　of　NM\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\{e_{m}\}_{m\in　{\mathbb　{N}}_{M}}$$\end{document}　is　not　necessarily　1Me2　pi　im　center　dot　Mm　is　an　element　of　NM\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\left\{　\frac{1}{\sqrt{M}}e＜^＞{\frac{2\pi　im\cdot　}{M}}\right\}　_{m\in　{\mathbb　{N}}_{M}}$$\end{document},　and　that　our　method　also　applies　to　phase　retrievability　in　CM\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$${\mathbb　{C}}＜^＞{M}$$\end{document}.　For　the　real　Hilbert　space　(HM,⟨center　dot,center　dot⟩)\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\big　({\mathbb　{H}}＜^＞{M},\,\langle　\cdot　,\,\cdot　\rangle　\big　)$$\end{document}　induced　by　(HM,(center　dot,center　dot))\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\big　({\mathbb　{H}}＜^＞{M},\,(\cdot　,\,\cdot　)\big　)$$\end{document},　we　present　a　sufficient　condition　on　phase　retrieval　frames　{emTng}m　is　an　element　of　N4M,n　is　an　element　of　NM\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\{e_{m}T_{n}g\}_{m\in　{\mathbb　{N}}_{4M},\,n\in　{\mathbb　{N}}_{M}}$$\end{document},　where　{em}m　is　an　element　of　N4M\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\{e_{m}\}_{m\in　{\mathbb　{N}}_{4M}}$$\end{document}　is　an　orthonormal　basis　for　(HM,⟨center　dot,center　dot⟩)\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\big　({\mathbb　{H}}＜^＞{M},\,\langle　\cdot　,\,\cdot　\rangle　\big　)$$\end{document}.　We　also　give　a　method　to　construct　and　verify　general　phase　retrieval　frames　for　(HM,⟨center　dot,center　dot⟩)\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\big　({\mathbb　{H}}＜^＞{M},\,\langle　\cdot　,\,\cdot　\rangle　\big　)$$\end{document}.　Finally,　some　examples　are　provided　to　illustrate　the　generality　of　our　theory.