Can someone explain how I might approximate the guitar tone in this clip?

Bob,
 
I would use a Fender Twin or Mesa Boogie amp, clean channel, with some pretty heavy compression and stereo chorus for the rhythm part.  It is compressed pretty heavily.  A lot of the effect is also produced by the droning open strings and his use of 2nd intervals...
 
Lead part he has added some delay and over-drive and backed off the chorus.
 
Looks like he is on the bridge pickup for the rhythm part and then switched to the Neck/Bridge position for his solo to add some more body to the sound.
 
This link may be helpful...
 
http://mesaboogie.com/news/2010/06/dominic-miller-sting-symphonicity-tour/
 
 

Thank you both SO much.  A friend of mine that I used to be in a band with also played a Mesa Boogie, and he had that same kind of sound - I loved it 30 years ago, and still do.
 
I will put the info to good use, 
 
Bob Bone
 

Thanks soooo much, guys.  Many moons ago my guitar player had a Mesa with that kind of sound, and it has haunted me ever since.
 
I will be moving some gear around in the studio today, and then I will sit down and craft up something based on the guidance you guys have provided above.  :)
 
Bob Bone
 

Don't discount the tone from a rosewood fingerboard either... little things add up... I notice in the lead he's changing pickups throughout as well...

robert_e_bone

sharke
Yeah those cluster chords go a long ways to making that sound. Any notes a tone or half tone apart sustained together will add to the chorus effect. I like the sound of cluster chords so much that I've stretched the hell out of my fingers playing them over the years

Good point - I happen to be 6'9" (played basketball in a bunch of countries, back in the day), and I have really big Holdsworth-sized hands.  When I used to play guitar on some songs in an original band I was in, I used to voice chords with REALLY wide intervals, as well as lots of 2nd's and such, where having big hands was essential.  There were chords I could play that most folks couldn't, and we had a pretty good following and guitar players used to try to figure out what in the world I was playing - because they couldn't reach the same notes.  Hee hee
 
Bob Bone
 

 
You learn something new about forum members every day. I had no idea you were a giant 
 
I'm only 5"10 but nonetheless I have hands like spades and pretty long fingers. Years of playing Bach fugues and other music not written for guitar has pretty much contorted and stretched my fingers into a state in which I can finger pretty much anything you throw at me, even if I have to use knuckles and sides of fingers etc. It's amazing how much your fretting fingers stretch - my left hand digits are a good half an inch longer than my right. However short, stubby fingers are no obstacle for a determined player, just look at the late Danny Gatton. 
 

Using your thumb, right?

robert_e_bone
I can go from a low E string 1st fret (F), to a high E string 7th fret.  Bwa ha ha.  :)
 
Too bad I play solos at the speed most people tune up.  (I am good with chords, but that's about it)
 
Bob Bone
 

I can just about do that, although my pinky is right in front of the fret. I can definitely go from 1st to 6th very comfortably though.