Does anything happen when a pawn reaches the opposite baseline? Can it take a baseline character...can it rescue a character out of a keep? Does it just stand there like an asshole until a spell can redirect it?
There is nothing in the rulebook that says anything special happens when a PIKE reaches the baseline, so at that point the piece is just taking up space.
The bottom of page 3 of the rulebook indicates that no unit on any baseline can be taken under normal circumstances, to unless a spell like FIRE STARTER is used, it is not even fodder.
There is not as much reason to move towards an enemies baseline as there is in chess. You can't normally take pieces there, and you get victory points by taking or occupying the capital, which usually happens outside of the baseline. For this reason, I imagine there was no need to worry about what happens if a PIKE reaches the baseline, because it is so unlikely to happen before the game ends.