Today() does generate timestamp, but it is in JMP date time value Using dates, times, datetimes and durations in JMP.
If you use AsDate() it will most likely have some characters which are not allowed in filenames. What type of format do you want? You can build it from the datetime Today() gives you, check out Scripting Index and Date Time part:
From there you can find functions like Year(), Month(), Day(), Hour(), Minute() and Second() with these (combined with Char() or Eval Insert() you can build most robust version of the timestamp you want.
Edit:
This is a bit of long code, but it should be fairly robust and basically give you YYYYMMDD_hhmmss formatted string
Names Default To Here(1);
cur_time = Today();
yyyy = char(Year(cur_time));
//add spanning
mm = char(Month(cur_time));
mm = Substr("00", 1, 2 - Length(mm)) || mm;
dd = char(Day(cur_time));
dd = Substr("00", 1, 2 - Length(dd)) || dd;
hh = char(Hour(cur_time));
hh = Substr("00", 1, 2 - Length(hh)) || hh;
min = char(Minute(cur_time));
min = Substr("00", 1, 2 - Length(min)) || min;
ss = char(Second(cur_time));
ss = Substr("00", 1, 2 - Length(ss)) || ss;
timestamp_str = yyyy || mm || dd ||"_"|| hh || min || ss;
-Jarmo