We construct a one-parameter family of generalized Mathieu functions, which are reduced quaternion-valued functions of a pair of real variables lying in an ellipse, and which we call λ-reduced quaternionic Mathieu functions. We prove that the λ-RQM functions, which are in the kernel of the Moisil-Teodorescu operator D +λ (D is the Dirac operator and λ∈R\{0}), form a complete orthogonal system in the Hilbert space of square-integrable λ-metamonogenic functions with respect to the L2-norm over confocal ellipses. Further, we introduce the zero-boundary λ-RQM-functions, which are λ-RQM functions whose scalar part vanishes on the boundary of the ellipse. The limiting values of the λ-RQM functions as the eccentricity of the ellipse tends to zero are expressed in terms of Bessel functions of the first kind and form a complete orthogonal system for λ-metamonogenic functions with respect to the L2-norm on the unit disk. A connection between the λ-RQM functions and the time-dependent solutions of the imaginary-time wave equation in the elliptical coordinate system is shown.