Euler’s identity e^{iθ} = cos θ + i sin θ expresses rotation in the complex plane. We extend this identity by replacing the imaginary unit i with an operator J satisfying J^2 = −I, where J acts on a doubled dynamical state encoding the four fundamental PDE universality classes: elliptic, hyperbolic, parabolic (diffusive), and unitary (Schrodinger-type). This gives the Generalized Euler Identity e^{θJ} = cos θ I + sin θ J , which performs Wick rotation through physical law itself. This formalism reveals a natural holofractal cyclicity among the canonical dynamical regimes.