The 1878-1881 period in NM was known for the Lincoln County War in 1878, and the Dodge City Gang which terrorized Las Vegas from 1879-1880. Several gunfighters from the gang headed out of town to Tombstone, Arizona after a vigilante party of townspeople was formed, though some returned to NM in the next few years. Billy the Kid, present for the Lincoln County War, was killed in Fort Sumner, NM by Sheriff Pat Garrett in 1881.
The number of gunfights in the US forms a neat bell curve--starting with 13 in the period from 1854-1859, peaking at 106 in 1875-1879, and dropping off to 9 by 1910-1914.
In 1908, for example, there were only four gunfights in the US--though two were in New Mexico. One of the remaining two was the fight between Bolivian soldiers and Butch Cassidy and the Sundance Kid. (This was counted as a US gunfight since two of the protagonists were Americans.)
Still, I guess it's all in what you count as a gunfight. An 18-year-old boy was shot to death by a 16-year-old in the parking lot of the Santa Fe Place mall this week--a fight over a girl apparently. So it's not quite over yet.
Illustration Credits and References
Statistsics on gunfighting appeared in Almanacs of American Life: Victorian America 1876-1913 by Crandall Shifflett (1996); the author credits Encyclopedia of Western Gunfighters by Bill O'Neal (1979) for this data.
Other information from the Legends of America website.
Photograph of Billy the Kid's death marker courtesy of chucksville.org