The Biot-Savart law can be derived from Maxwell's equations for 3D using the vector product (via the curl). On the other hand, in a purely two-dimensional (2D) world, the vector product is not well defined and, therefore, its derivation is not mathematically feasible. For this situation, despite some efforts, it seems convenient to formulate the problem using Geometric Algebra.
I would be interested in knowing the formulation of the Biot-Savart law in 2D to obtain the magnetic field at a generic point. According to my intuition, the magnetic field should be a bivector, but I'm not sure.
