Why are absolute values more natural than discrete valuations? It is true that considering the archimedean places as well is more general, but that still doesn't explain why it is more natural. If we consider both the definitions of an absolute value and that of a discrete valuation, they both seem arbitrary. However, discrete valuations have the extra appeal of being related to geometric objects (they arise as local rings at non-singular codimension 1 subschemes of integral schemes, and in various other situations: local rings at non-singular codimension 1 subschemes of a scheme birational to the original scheme, or more abstract situations like Zariski-Riemann spaces). It seems (to me at least) that the archimedean places don't enjoy a similar geometric analogy.
This leads me to ask: why are the archimedean places natural? Why do they always feel like they're "missing points" of rings of integers?
I should note in the interest of full disclosure that I'm unfamiliar with the ways of Arakelov theory.
 A: Perhaps the main reason why people care about archimedean places is that many results in algebraic number theory become much neater if you treat them on equal footing with finite primes, even before you get to Arakelov theory. E.g. zeta- and L-functions have contributions from infinite primes, Dirichlet's S-unit theorem is nicer to state, theorems in class field theory become nicer, and the whole adelic business becomes possible. As a concrete example, Kevin Buzzard's answer and especially Matt Emerton's comment in
Positive polynomial having roots modulo any integer
relies on the fact that you can treat complex conjugation like any other conjugacy class of Frobenius elements. 
On the other hand, when there is no need to care about infinite places, many (most?) people will definitely state what they want in terms of discrete valuations rather than absolute values, because, as you say, they are nicer and more intuitive. I don't think many will say "because $4\in{\mathbb Z}[i]$ has $(1+i)$-adic absolute value $1/16$...", they will say "because $4\in{\mathbb Z}[i]$ has valuation $4$ at $(1+i)$". 
