circular light followed my car..appeared within seconds beside my car (passenger side), then re-appeared in rearview mirror.

This event happened years ago.  
I was traveling solo in my car with a few personal belongings from Colorado to California.  It was very late at night and I was traveling through Arizona...I believe it would have been highway 40.  
I remember seeing a bright circular light in my rearview mirror..it followed me steadily for a very short while.  When I took my eyes off the mirror to look at the highway ahead...I glanced to my right and noticed the same size/intensity white light..facing me...at about the same speed I was traveling.  I could not determine the light's distance from my car...but it traveled in a very smooth/steady manner...and the light faced me for a short time (as I repeatedly looked out the passenger window.  
Seconds later...as I looked up in my rearview mirror...it appeared once again, behind me.  I remember being quite tired that night, as I was trying to drive non-stop from Colorado to California.  I distinctly remember thinking that something was observing me during that stretch of the highway.  I don't clearly remember how long after I saw this light that I experienced an electrical failure, but I remember driving into a small town and parking next to a gas station (it was closed for the night).  I got as comfortable as I could in the front seat...and pretty much slept sitting up in my car (as I had personal belongings surrounding me...to include items in the passenger seat.  
In the morning, I had either my generator or alternator replaced.  (I was driving a 67-Chevelle..and don't remember if it was the alternator or generator).
Witness indicates that the date of the sighting is approximate.  PD