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It is possible to solve the issue on the mathematical side.

$u=\sqrt{-1}$ has minimal polynomial over $\Bbb F_p$ equal to $(U^2 + 1)$.

We know $v^2 + u+1$ is zero in $\Bbb F_p[u]$ so let us multiply with the (Galois) conjugate $$0=(v^2+u+1)(v^2-u+1)=(v^2+1)^2-u^2=(v^4+2v^2+1)-(-1)\ .$$ The element $v$ has minimal polynomial $V^4+2V^2+2$ over the base field $\Bbb F_p$.

It remains to let sage make its choice for $w=\pm\sqrt v$, which has minimal polynomial $W^8+2W^4+2$. All you need is (using the default x or explicitly defining it as polynomial ring indeterminate):

sage: p = 0x1280f73ff3476f313824e31d47012a0056e84f8d122131bb3be6c0f1f3975444a48ae43af6e082acd9cd30394f4736daf68367a5513170ee0a578fdf721a4a48ac3edc154e6565912b
sage: K.<w> = GF(p^8, modulus=x^8 + 2*x^4 + 2)
sage: w.minpoly()
x^8 + 2*x^4 + 2
sage: v = -w^2
sage: u = -(v^2 + 1)
sage: u^2
4576545538729420598762745822889397370509838601207708465545582186285824315458656151272834027217178198654229063318759931344008864619718319130560845441720114764111976549023322410
sage: u^2 == -1
True

Then work with $u,v,w\in K$ as usual.