I've been wondering how the ending of Chrono Cross is truly resolved. This thread will speculate on a variety of inquiries; firstly, we shall begin with Serge.
Forced-Repression
We know that time travelers keep memory of the timeline that existed prior to its changing; those changed do not due to their timeline being new, and the prior one discarded, seen in the axiom How Changing the Past Affects Those in the Present and Future. When Serge used the Chrono Cross, he unified the dimensions, which would be undeniably changing time (history of 10 years). After his party defeats the Time Devourer and releases Schala, he and the party are sent in time to instance when Serge was originally called across the dimensions to Another World. Serge is told that he will lose all memory of the events of Cross; yet, since he is in the Darkness Beyond Time, which is outside of the normal flow, during the unification of the dimensions and altering of history, he should retain these memories; on Opassa Beach, he does mention FATE and Terra Tower.
Thus, I hold that Serge's memory is specifically repressed by Schala or some other mechanism, in order to allow him to live a normal life. Schala herself states, "You will lose all memory of this whole adventure and return to your own time;" thus, she may be its originator.
Time Bastard
The dimensions are unified so that Serge is born and lives; thus, would not the Serge returning from the Darkness Beyond Time encounter the Ideal Timeline's Serge?
GrayLensman has proposed that the Ideal Serge, who inexorably has no memories of the event as he was a product of the adventure and not its partaker, is discarded into the Darkness Beyond Time during the brief intermission between Serge passing out on Opassa and reawakening afterward in the Ideal Timeline.
And what of the party?
What happens to the party? Since only two members may accompany Serge to the Darkness Beyond Time, it is reasonable that all others are simply subject to the axiom How Changing the Past Affects Those in the Present and Future. The two present may have been transplanted into the Ideal Timeline as suggested above, or may have been discarded into the Darkness Beyond Time entirely; there simply is no way of telling since we cannot observe whether they also have fragmentary recollections of the adventure.