Deterministic Büchi + its complement covers LTL?
From your example, it is easy to derive a language $L$, such that neither $L$ nor its complement is recognized by a DBA.

Take an alphabet of four letter $\{a,b,c,d\}$, and let $L=((a+b)^* a^\omega)+(c^*d)^\omega$. It is not DBA-recognizable because of the $\{a,b\}$-part, and its complement is not because of the $\{c,d\}$-part.

For a less artificial example, you can for instance take on alphabet $\{a,b,c\}$ the language $L$ of words with infinitely many $b$ and finitely many $c$. Both $L$ and its complement have to check that the number of something is finite, so both are not DBA-recognizable.