In order to determine the splitting field of $t^{4}+2\in \mathbb{Z}_{3}[t]$, I first "guessed" the roots $1$ and $2$ then, by polynomial division, obtained
$t^{4}+2=(t^{2}+1)(t+2)(t+1)$
Since for $t^{2}+1$ to be $0$ I'd need $\sqrt{2}$, I came to the conclusion that $\mathbb{Z}_{3}(\sqrt{2})$ is the spitting field of $f$ over $\mathbb{Z}_{3}$
Is my idea correct ?
Your approach is correct, but I don't think that the expression $\sqrt2$ is used in this context. I would say that the splitting field of $t^4+2$ is $\mathbb{Z}_3[t]/(t^2+1)$ or that it is$$\{a+bs\,|\,a,b\in\mathbb{Z}_3\},$$where $s^2=2.$