In a world with two time dimensions there would be no such thing as causality: with a, b \in R^2 (the field of real numbers), a<b doesn't make sense, because two events can have the same distance from (0,0) but different angle \phi (modulo orientation of chosen frame of reference), and therefore I wouldn't be able to distinguish between past and future events in general. Causality breaks down.
Propagating along a 1D trajectory (ie flow of time) would then be along a 2D "trajectory" through which I would experience indefinitively many events at once, unclear which one impacts on which others. But clearly I perceive me writing this post "right now", and am about to push the send button..