<p>We extend dual quaternions to <i>n</i>-quaternions with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n\ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and investigate the solution to the matrix equation <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(AX=B\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>X</mi> <mo>=</mo> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> over <i>n</i>-quaternions. We obtain a necessary and sufficient condition for the solvability of the generalized Sylvester matrix equation <Equation ID="Equ9"> <EquationSource Format="TEX">\(AX+EXF=CY+D\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>A</mi> <mi>X</mi> <mo>+</mo> <mi>E</mi> <mi>X</mi> <mi>F</mi> <mo>=</mo> <mi>C</mi> <mi>Y</mi> <mo>+</mo> <mi>D</mi> </mrow> </math></EquationSource> </Equation>over <i>n</i>-quaternions, and provide a general expression for its solutions when it is consistent. As an application, we develop an encryption-decryption scheme for color images based on the latter equation, with experimental results demonstrating the scheme’s strong feasibility.</p>