After reading up on complex logarithms, I think I may have figured it out.
Complex logarithms let me rewrite
$$x^y$$
as
$$e^{y \ln (x)}$$
$\ln (x)$ is still a complex number (actually many of them). The real and imaginary parts of this number are $\ln (|x|)$ (which is now just a standard natural logarithm of a positive real) and $\arg (x)$ (still a multivalued function) respectively.
That lets me write my expression as
$$e^{y (\ln (|x|) + i \arg (x))}$$
which I can further expand to
$$e^{y \ln (|x|)} e^{i y \arg (x)}$$
Referring to the identity suggested by @blakedylanmusic, I can observe that
$$r = e^{y \ln (|x|)}$$
$$\theta = y \arg(x)$$
Now applying the identity, I rewrite the expression as
$$e^{y \ln (|x|)} (\cos (y \arg (x)) + i \sin (y \arg (x)))$$
Now
$$a = e^{y \ln (|x|)} \cos (y \arg (x))$$
$$b = e^{y \ln (|x|)} \sin (y \arg (x))$$
If I understand the multiplicity of complex logarithm correctly, the multiplicity of the complex logarithm is captured by $\arg(x)$ returning many possible angles. That multiplicity is then collapsed away when we pass that value back into $\sin ()$ and $\cos ()$, so this is expression actually produces a unique complex number - the exact result of the initial exponentiation.
