Is it appropriate to apply Euclidean Distance to Complex Numbers? Would complex numbers be considered as part of Euclidean Space? Would this measurement give an accurate result? If not, what would be a more appropriate distance measurement/similarity measure with respect to complex numbers?

Euclidean distance is exactly what we get when we consider the modulus of a complex number: $$|a+bi| = \sqrt{a^2 + b^2}$$
so it makes complete sense to use Euclidian distance, and this is in some sense the most natural distance to choose, since $\mathbb C$ is isomorphic to $\mathbb R^2$.
