Total linkage of quaternion algebras and Pfister forms in characteristic two

We study the subfields of quaternion algebras that are quadratic extensions of their center in characteristic 2. We provide examples of the following: two non-isomorphic quaternion algebras that share all their quadratic subfields, two quaternion algebras that share all their inseparable but not all their separable quadratic subfields and two quaternion algebras that share all their separable but not all their inseparable quadratic subfields. We also discuss quaternion algebras over global fields and fields of Laurent series over a perfect field of characteristic 2 and show that the quaternion algebras over these fields are determined by their separable quadratic subfields. Throughout, these linkage questions are treated in the more general setting by considering the linkage of Pfister forms.