  Quaternion extensions are often the smallest extensions to exhibit special
properties. In the setting of the Hasse-Arf Theorem, for instance, quaternion
extensions are used to illustrate the fact that upper ramification numbers need
not be integers. These extensions play a similar role in Galois module
structure. To better understand these examples, we catalog the ramification
filtrations that are possible in totally ramified extensions of dyadic number
fields. Interestingly, we find that the catalog depends, for sharp lower
bounds, upon the refined ramification filtration, which is associated with the
biquatratic subfield. Moreover these examples, as counter-examples to the
conclusion of Hasse-Arf, occur only when the refined filtration is, in two
different ways, extreme.
