Phasors, from what I can understand, are a way of representing the addition of and scalar multiplication of sine waves with constant frequency (so that only phase and amplitude differ), and this is accomplished by using a complex number, and representing amplitude as the magnitude of the number, and phase as the argument of the number.
Has anyone generalized this concept to two argument waves like $f(x,y) = sin(x) + sin(y)$? For example, here the two dimensional sinusoid $g(x,y) = A\sin(x+\phi)+ B\sin(y+\psi)$ might be represented by $(\phi,\psi,A,B)$. The fact this is a 4-tuple makes me immediately want to try a quaternion representation to define an algebra for functions of the same form as $g$. Has this been done before, and does it admit a generalization to $n$ dimensions?
Phasors are useful for linear systems (as far as I've used them). This means you can consider each term separately, do each transformation separately, and at the end sum them together.