Quaternion algebra \(\mathbb {H}\) as a generalization of complex numbers is a noncommutative associative algebra. Recently, quaternionic Fourier analysis has interested some mathematicians due to its potentials in signal analysis and color image processing. This paper addresses quaternionic generalized affine phase retrieval (QGAPR) in quaternion Euclidean spaces \(\mathbb {H}^{M}\) . Let \((\mathcal {F},\,{h})=\{(F_{n},\,h_{n})\}_{n\in \mathbb {N}_{N}} \subset \mathcal {M}_{M\times L}(\mathbb {H})\times \mathbb {H}^{L}\) , where \(\mathcal {M}_{M\times L}(\mathbb {H})\) denotes the set of all \(M\times L\) quaternionic matrices. We introduce the concept of QGAPR in \(\mathbb {H}^{M}\) associated with \((\mathcal {F},\,{h})\) which aims to recover signals exactly from the magnitude of their quaternionic affine transformations, and establish the characterizations of QGAPR for \(\mathbb {H}^{M}\) based on real Jacobian matrices and phaselift operators respectively. Also, we prove that is sufficient for the existence of \((\mathcal {F},\,{h})\) allowing QGAPR for \(\mathbb {H}^{M}\) , and that, under a mild condition, is also necessary for the existence of \((\mathcal {F},\,{h})\) allowing QGAPR for \(\mathbb {H}^{M}\) . Finally, we prove that, if , the set of all \((\mathcal {F},\,{h})\) allowing QGAPR for \(\mathbb {H}^{M}\) is not an open set, and its every point is not an isolated point. This makes QGAPR very different from quaternionic generalized phase retrieval (QGPR).
