Williams Lake from a few years back. Williams Lake is a natural alpine lake which does not have fish because it freezes in winter. It is near Wheeler Peak, the highest point in New Mexico. Wheeler Peak is on the ridge line east of the lake and can be reached by climbing the talus slope. Technical rock climbing gear is not needed, but the slope is very steep and very rocky. Several avalanche chutes are apparent along the trail, winter travelers should be alert to these hazards, and debris may be present in the spring and summer. The lake's name is in reference to William Frazer, a gold miner who staked claims in the area and co-founded the mining camp of Twining during the late 1800s.