<p>Complex fuzzy matrix equations serve as a powerful tool for addressing problems in imprecise and complex systems. However, conventional methods can only yield a single strong (or weak) fuzzy solution or fail to obtain an exact solution. Although the theory of generalized inverse has been employed for efficient solving, constructing the general fuzzy algebraic solutions for complex fuzzy matrix equations based on different generalized inverses remains a key challenge. To address this, this paper introduces the CMP inverse and investigates its key properties. Then, based on this, for the first time, derives the general fuzzy algebraic solutions for the square complex fuzzy matrix equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(A\widetilde{Z}=\widetilde{B}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mover accent="true"> <mi>Z</mi> <mo stretchy="true">~</mo> </mover> <mo>=</mo> <mover accent="true"> <mi>B</mi> <mo stretchy="true">~</mo> </mover> </mrow> </math></EquationSource> </InlineEquation>. The study not only presents a necessary and sufficient condition for the existence of general fuzzy algebraic solutions to <i>R</i>-consistent complex fuzzy matrix equations and develops a corresponding algorithm, but also extends the theory to general fuzzy least-squares solutions for inconsistent equations. Furthermore, as an extension of the theoretical application, we extend the theory to find the common solutions to the coupled equations <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(A\widetilde{Z}=\widetilde{C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mover accent="true"> <mi>Z</mi> <mo stretchy="true">~</mo> </mover> <mo>=</mo> <mover accent="true"> <mi>C</mi> <mo stretchy="true">~</mo> </mover> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\widetilde{Z}B=\widetilde{D}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>Z</mi> <mo stretchy="true">~</mo> </mover> <mi>B</mi> <mo>=</mo> <mover accent="true"> <mi>D</mi> <mo stretchy="true">~</mo> </mover> </mrow> </math></EquationSource> </InlineEquation>. Finally, a comparative numerical analysis with existing methods is included, demonstrating the efficiency and universality of the proposed methods.</p>