Canyons are inherently risky. Flash floods occur without notice on sunny days. Technical skills & surrounding topography knowledge required yet does not eliminate risk.
If we'd just turned around sooner!
Beeman Canyon lies on the far northern end of the frontal range, the last drainage before the major road cut heading up the mountain to Cloudcroft. It starts out as a nondescript wash hike, bisected numerous times by Jeep and ATV tracks, in fact, you may find it just as desirable to hike the old Jeep track as the lower wash drainage... your choice on that. It is after the Jeep track dies off up and to the left that the real fun begins. Now, you move easily up through the wash. The rocky cliffs draw closer in, and you finally reach the Portal... a narrowing created by two large, distinctive rocks balanced on the tilted slabs of the drainage bottom... this begins the section called the Jumble... about a mile of basketball, volkswagon, school bus and house size boulders tumbled into and filling the canyon bottom. It's a tumbled maze... and you get to work your way up and through, up and over, up and around... all challenging... all pure fun. The drainage is narrow, so the route finding is minor... basically, you simply continue to go up!
This draws to a distinctive close at a ledge waterfall... rotten rock, so instead of climbing it you may choose to work your way around on the left... shrubby, but doable.
From this point on you are back into a basic wash, with a much flatter grade for awhile... then you occasionally get to enjoy some bedrock walking, in an every narrowing defile. Eventually you come to a distinctive split, the forks (called that for sure because someone has painted "The Forks" on the rock there!) and your path leads to the right. I have not hiked the left (north) fork, but the map shows that if you go that way you will reach the top of the ridge up at the meadows area below Winter Peak... and if you plan to loop down Dry Canyon and back around to your car and desire a shorter and easier route, that left turn may be for you. Otherwise, continue on up the ever steeper right branch. As you near the top, the ridge to your right is really the edge of the cliffs overlooking the Tularosa Valley. You have the option of staying down in the drainage all the way to the very top, or angling to the right to walk the ridge above the drainage. Either choice will keep you moving on up to the top...
...where you will first reach the launching lip for the local hang gliders... and the jeep track that accesses the mountain. Follow it a little ways further up and you will be at the top of Winter Peak... and have hiked the full extent of the drainage of Beeman Canyon. At this point you have the option of falling the Jeep track down to Dry Canyon wash for a nice loop hike back around to the mouth of Beeman... or, head back the way you came... or get creative... Many folk will likely simply opt to turn around at the Waterfall above the Jumble, vesting their time in the most fun section of the drainage. Not a bad idea.
WARNING! Hiking and outdoor related sports can be dangerous. Be responsible and prepare for the trip. Study the area you are entering and plan accordingly. Dress for the current and unexpected weather changes. Take plenty of water. Never go alone. Make an itinerary with your plan(s), route(s), destination(s) and expected return time. Give your itinerary to trusted family and/or friends.