  We classify all complex quadratic number fields with 2-class group of type
(2,2^m) whose Hilbert 2-class fields have class groups of 2-rank equal to 2.
These fields all have 2-class field tower of length 2. We still don't know
examples of fields with 2-class field tower of length 3, but the smallest
candidate is the field with discriminant -1015.
