We give some topological characteristics of the coamoeba of a generic k-dimensional affine space and two stronger versions, specific for the affine case, of a result by Nisse, Sottile and the author. We also give topological and partly algebraical characterizations of the amoeba and coamoeba in some special cases: k=n-1, k=1 and, when n is even, k=n/2, in the last case with a certain emphasis on the example n=4.