Question

You can solve the differential equation y''+y=0 using complex numbers. Just write $$(\partial^2 + 1) y = (\partial +i)(\partial -i) y$$ and you are now dealing with two order one differential equations that are easily solved $$(\partial +i) z =0,\qquad (\partial -i)y =z$$ The multivariate case is a bit harder and uses quaternions or Clifford algebras. This was done by Dirac for the Schrodinger equation ($-\Delta \psi = i\partial_t \psi$), and that led him to the prediction of the existence of antiparticles (and to the Nobel prize).