Work in the context of combinatorial games as introduced by Conway. For surreals, the definition of the product is forced by the requirement that surreals should form an ordered field. Say, if $s' < s$ and $t'< t$, then we should have $s-s'>0$, $t-t'>0$, hence $(s-s')(t-t') >0$ and $st > st' + s't - s't'$. In game theoretical terms, if $s'$ is a left option of $s$ and $t'$ is a left option of $t$, then $st' + s't - s't'$ should be a left option of $st$. The argument is similar for the other options defining $st$.
However, the class of general games is not linearly ordered, and it is not necessarily the case that if $g'$ is a left option of a game $g$, then $g'< g$. So we have more possibilities for the definition of the product. For example, we can take as left options of $gh$ only the games of the form $gh' + g'h - g'h'$, with the further assumption that not only $g'$, resp $h'$, are left options of $g$, resp $h$, but also $g' < g$ and $h' < h$. Of course, the other conditions should be treated similarly. It seems that this alternative definition of the product leaves unchanged the product of two surreals and trivializes the product of many games wich are not (surreal) numbers. Thus probably the above definition is not very interesting.
Are there more interesting intermediate possibilities? In particular can we define a product on the class of all games in such a way that (products of surreals remain unchanged and) we get a game which is an imaginarry unit, that is $g^2=-1$? Perhaps the answer is no, but maybe it is worth trying.