The anomalous Hall effect (AHE) with multi-degree-of-freedom coupling has become a research hot spot in the condensed matter field due to its multiple tunable states, and exploring effective tuning methods for spin, valley, and layer degrees of freedom holds significant research value. Through the first-principles calculations and symmetry analysis, we investigate the properties of bilayer altermagnetic (AM) material $\mathrm{Ca}{\left(\mathrm{CoN}\right)}_2$, which is protected by $S_{4z}$ symmetry, with a particular focus on the influence of interlayer coupling and sliding on the band structure. For bilayer $\mathrm{Ca}{\left(\mathrm{CoN}\right)}_2$ with interlayer antiferromagnetic (AFM) coupling, the interlayer opposite spin sublattices can be connected by symmetries such as $S_{4z}$, classifying it as an AM system. In contrast, systems with interlayer ferromagnetic (FM) coupling belong to the fully compensated ferrimagnetism category. Interlayer sliding breaks the $S_{4z}$ symmetry of the system, thereby controlling the coupling of spin, valley, and layer degrees of freedom. Sliding along the $x$ or $y$ axis breaks the $S_{4z}$ symmetry, inducing opposite valley polarization and layer polarization, which further leads to layer-locked Berry curvature and spin-valley-layer polarized AHE. Bilayer systems with interlayer AFM coupling can switch the spin, valley, and layer degrees of freedom via sliding, while those with interlayer FM coupling can only switch the valley and layer degrees of freedom. This work reveals the sliding rules of bilayer AM systems protected by $S_{4z}$ symmetry and expands the implementation methods for AHE with multi–degree of freedom tunability.

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• Anomalous Hall effect
• Ferroelectricity
• Magnetic coupling
• Spin-orbit coupling
• Spintronics
• Structural properties
• Valley degrees of freedom
• Valleytronics
• 2-dimensional systems
• Altermagnets
• Bilayer films
• Layered crystals
• First-principles calculations
• Wannier function methods