The ultimate frontier of the particle physics beyond the Standard Model is the research of axion-like particles, to explain the absence of CP violations in the strong interaction and provide dark matter candidates. In the MADMAX (MAgnetized Disc and Mirror AXion) experiment, the principle of axions detection relies on their predicted emission of an electric field proportional to the square of a static magnetic field using magnetized dielectric disks. In this experiment, the magnetic field would be provided by a 9 T NbTi dipole magnet, 6.0 m long and with an aperture diameter of 1.3 m, producing a Figure of Merit over the disk surfaces of 100 T²m² along 2 meters, enough to detect axions with masses in the range of 100 μeV. This paper reports the 2D and 3D conceptual magnetic design of the MADMAX dipole. Starting from a general overview of the best magnet configuration, it introduces the parameters and design methodology to adapt the dipole to the experiment constraints. Then, the 2D and 3D designs are described, focusing in particular on the cross-section, the harmonics optimization in 2D and the heads optimization in 3D with and without iron, made by coupling a dedicated analytical approach to genetic algorithms. In the end, a section shows the optimization of the conductor cost as a function of the overall current density.