  We study the rich behavior of ergodicity and conservativity of Cartesian
products of infinite measure preserving transformations. A class of
transformations is constructed such that for any subset $R\subset \mathbb Q\cap
(0,1)$ there exists $T$ in this class such that $T^p\times T^q$ is ergodic if
and only if $\frac{p}{q} \in R$. This contrasts with the finite measure
preserving case where $T^p\times T^q$ is ergodic for all nonzero $p$ and $q$ if
and only if $T\times T$ is ergodic. We also show that our class is rich in the
behavior of conservative products.
  For each positive integer $k$, a family of rank-one infinite measure
preserving transformations is constructed which have ergodic index $k$, but
infinite conservative index.
