Why generalize the Euclidean metric? It is well known that the Euclidean metric can be generalized to $\Bbb R^n$ by $\sqrt{(x_1-x'_1)^2+\cdots + (x_n-x'_n)^2}$, and that under this generalization it is still a metric and satisfies various other properties of the two and three dimensional cases.
But other than as a mathematical curiosity, why do this? Does this metric appear naturally in any mathematical problems or fields of study? Or is the only reason to generalize because we can?
 A: There are probably deeper answers to this, but one that comes to mind is that just because our world (high level physics considerations aside) is three dimensional, that doesn't mean that the only things that we want to measure in it are three dimensional. As an example let's assume you have a data set, say a point for $N$ different people and associated to each person you have a number $R$ of numerically-valued characteristics, e.g. height in centimeters, weight in kilograms etc. Then to each person we can associate a point in $R$ dimensional space, whose coordinates are the values of the characteristics. 
Given this set of data we'd like to be able to tell whether two people are 'close' in some sense, so we need a metric, a notion of distance between the points. We already have the Euclidean metric in 3 dimensions, and we can generalise it to $R$ dimensions, so that's one notion of distance that we can choose. There are other metrics of course, many better suited, but this is an example of a situation where we need a metric on a space with more than 3 dimensions to tell us something about the real world.
A: One important reason is just because we can. More than that, spaces of $n$ dimensions appear constantly in other fields of study like physics (even when they do not represent the physical space we live in, which seems to be 3 dimensional), and having that metric is usefull. For example phase spaces of a mechanical system have $2n$ dimensions, being $n$ the degrees of freedom. In quantum mechanics arbritarily high dimensions appear as Hilbert spaces are used, even infinite ones. Having a normed space is usefull when leading with these things.
More than that, we do it because we can, and I think that's the more important reason if we're talking about math. Also in math it leads to deeper results in other areas. There are more strange metric spaces with more strange distance functions that would ask about. But most of them are applied to something.
