The theory of scale relativity provides a new insight into the origin of fundamental laws in physics. Its application to microphysics allows us to recover quantum mechanics as mechanics on a non-differentiable (fractal) spacetime. The Schrödinger and Klein–Gordon equations are demonstrated as geodesic equations in this framework. A development of the intrinsic properties of this theory, using the mathematical tool of Hamilton's bi-quaternions, leads us to a derivation of the Dirac equation within the scale-relativity paradigm. The complex form of the wavefunction in the Schrödinger and Klein–Gordon equations follows from the non-differentiability of the geometry, since it involves a breaking of the invariance under the reflection symmetry on the (proper) time differential element (ds ↔ −ds). This mechanism is generalized for obtaining the bi-quaternionic nature of the Dirac spinor by adding a further symmetry breaking due to non-differentiability, namely the differential coordinate reflection symmetry (dx^μ ↔ −dx^μ) and by requiring invariance under the parity and time inversion. The Pauli equation is recovered as a non-motion-relativistic approximation of the Dirac equation.