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?

You can certainly use the Euclidean idea of distance between two complex numbers
$$
|z_2 - z_1| = \sqrt{(x_2-x_1)^2 + (y_2 - y_1)^2}
$$
However, the complex plane contains more algebraic structure than plain euclidean space $R^2$ so calling complex numbers "part of the euclidean space" is not quite accurate. Here is a link to another question with a discussion of the differences between the two.
