Over 30 min, observed a starlike object, moving rapidly in the vertical, and horizontal, varying intensity, for 5-10 seconds each time.

Delta 8934 MIA-MSY, B757, flight level 380, we began to observe a small, point-light object moving rapidly in the vertical and horizontal. The sighting was towards our 11 o’clock and we were headed due West, over the Gulf of Mexico, offshore from KSRQ, approximately between BAGGS and REMIS waypoints.
It was a very dark night, and we had the lighting as dim as possible in order to help us detect thunderstorm build ups over the Gulf. The object appeared intermittently over about 30 minutes, and would vary in brightness intensity. Sometimes the object glowed very bright, much brighter than a typical planet. Then it would fade to a dim point, and accelerate in various directions, unlike most aircraft I’ve ever seen, including military fighter aircraft I have flown. It would appear for about 5-10 seconds each time. We observed it illuminate, and then accelerate rapidly straight down. It illuminated, and went pure vertical. It got very bright, then dim, and accelerated horizontally, and continued into an arcing zoom climb before fading out. A nearby shooting star occurred, which looked very distinct from this phenomena. Various iterations of these maneuvers occurred and the sighting was reported to ATC. ATC confirmed there were no unusual aircraft out tonight besides the usual traffic. At first sighting, the object appeared in the general direction of another airliner at 40 miles to our 11 o’clock, and had a very distinct appearance from strobed aircraft lighting. ATC then said there was a previous report in that area 2 hours prior.
We turned northwest towards HRV (New Orleans area) and lost sight of any further activity.