Computations in extended finite field p^2
Generally, an element of $\mathbf F_{p^2}$ can be represented as $a+bu$ with $a,b \in \mathbf F_p$ and $u$ a root of an irreducible polynomial of degree $2$ over $\mathbf F_p$. If this polyomial is $X^2+1$, then we have $u^2 = -1$, and in that case it is customary to denote it by $i$ (in an analogy with complex numbers). There is nothing that mandates the use of the polynomial $X^2+1$, however. In your case, PARI/GP uses the polynomial returned by ffinit(821^2), which is the polynomial $X^2+X+1$. Thus the relation to use in your computations is $u^2 = -(u+1)$, and not $u^2 = -1$.

Now, can you force PARI/GP to use the polynomial $X^2+1$? Yes, but remember that the polynomial must be irreducible, and $X^2+1$ is irreducible over $\mathbf F_p$ if and only if $p \equiv 3 \pmod 4$ (prove this!). So it will not work modulo $821$, but would work for example modulo $823$. You do it by providing the polynomial, instead of just the order, to ffgen: u = ffgen(Mod(1,823)*u^2 + Mod(1,823), 'u).

Also, although in this case the primitive $3$rd root of unity $z$ equals $u$, note that it is not true in general.