It may not be the alternator... but most likely is.
My Stylus does the exact same thing and I have replaced the alternator, with a factory unit.
Mine seems to happen mostly when the outside fans kick on. as soon as the shut off, the lights go out... I've also noticed the do it at high RPM hard accelerations.
I think I had a bad ground or something, somewhere.
It has not affected the running or drivability of the car, so I haven't looked any further... maybe one day I will.

1995 Trooper LS auto 3.2 DOHC /w SOHC intake
1989 Trooper 2.6 auto
1989 I-Mark RS DOHC 1.6
1991 Stylus XS DOHC 1.8