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Goose Lake is gorgeous - one of the prettiest places in New Mexico. It's about 7 miles up from Red River, and takes 1.5 to 2 hours, on a road that is rugged. You need good ground clearence and 4WD. The road is single track so you will have to back down or pull off if someone is heading in the opposite direction. It's worth the trip up, and you can rent ATVs in Red River if you don't want to drive your own car/truck/suv.
The area is full of tourists from about 9 am to 6 pm, but you can camp right on the lake and when all the tourists leave it is quiet and dreamy. You may want to pack light because you have to haul your gear to the campsites and at almost 12,000 feet altitude that's not an easy task. Few travelers are up to the task, so you could find yourself with very little company.
You can fish for rainbow trout and there are Big Horn Sheep not mountain goats around (particularly at dusk). If you are up to it, take the switch back trail up and over the top at day light and see the elk.