I went out to barn to check on animals. It was very dark, some starlight, this is a very remote area with limited ground lights. There were two blue white lights, arranged like headlights but much larger, floating slowly out of the canyon and into the valley, at about 300 ft. elevation, and maybe 300 yds. away, seeming to follow the road. The lights blinked occasionally, there were no port/starboard lights or strobes. I was thinking maybe it was a heliocopter, so I turned off the engine of my car, but there was no discernible noise, other than wind. It was very, very windy. The lights moved closer, and I would estimate that they moved at about 10-15 mph. They stayed parallel and in the same plane. I pulled out my binocculars (which I keep in the car for observing my cattle, etc.), but couldn't make out any shape or profile. As the lights passed behind my house, I reversed the car and decided to follow it. The lights proceeded to the South-Southeast slowly, and when I pulled out of our driveway, I was parallel to it, and pulled into the road to pursue it. It was to the East of my location, still moving very slowly and very low, over the hills. I kept my speed at about 25 mph, thinking that I could intercept it out on the highway. And I thought to myself, "And then what?" and began to be a little intimidated. After I had traveled about .30 mile on the dirt road, the lights veered directly East and disappeared behind a low hill. I have lived here for 30 yrs. and figured out about where I might intercept it, so I went to the highway (NM Hwy 78) and proceeded East for about 4.5 miles, but didn't see it any longer. I drove up to a higher vantage point, but didn't see it anywhere.I felt very nervous suddenly and started to tremble. I had had another "encounter" in 1992, but this time, I feel a lot more intimidated, and though the curiosity and desire to pursue is there, I am quite uncomfortable. I am very tired, but not sure I can sleep tonight. As curious as I am, I wish this hadn't happened. I hope all my cows are there in the morning....