Question: When playing guitar,especially improvising,is it best to think of the notes a actual lettered notes or Scale degrees related to a tonic ? Since the keys are easily movable , I'm thinking that its better just to know the degree you want to play and knowing where that is related to the tonic.

Answer: 
Every guitar player is different so it depends. I have a friend who took piano lessons and read music before learning guitar and she would much rather know the names of the notes. 

I play based on sound and feel and knowing where to put my fingers relative to where they are now to get to the note I want next relative to the note I am on. 

But I have a musical ear. If you want me to play something, I would rather hear it than see it on tab. If you give me standard notation and ask me to play what is written I will get back to you in a few days. If I hear it, I can play it back immediately. 

I personally have no need to know the names of the notes I am playing while I am playing them (I **could** tell you if I **thought** about it). I just need to know how to make what I hear in my brain translate to my instrument and the easiest way for me to do that is to start on the right note and go from there. 

I am quite certain that if I was improvising and I tried to **engage the part of my brain that would allow me to identify the notes** I was playing, it would stifle my creativity and instinct. I think the best improvisers play by **allowing their mind to engage with their soul and play what "feels" right. Thinking about the NAME of the note they wanted to play next - would short circuit that connection.** Who cares what the note is called as long as it sounds good!

If you can improvise without knowing the **names** of the notes you are playing, your time will be better spent improving your craft by playing more - than by **trying to learn to identify each note you are playing while improvising.** That is my assessment based on my own personal playing style. Your mileage may vary.
It probably depends on how you 'hear' music.

For example ...

Absolute pitch :

I have recently been playing with a folk vocalist who doen'st play an instrument so we spent a bit of time getting the song into the right key for her. Her first choice was always pretty much the orignal key of a recording she'd learnt from - any other key felt 'odd', implying she might have something like perfect pitch. I think at that stage at least she's hearing music kind of in absolute terms : eg a C is a C , not a D even if you transpose. If your'e like this then perhaps it's more comfortable to work out the individual notes and get to know the fretboard that way. That's a lot to remeber though.

Relative to Tonic :

If you're able to hear the interval between the tonic and the notes of a tune (eg it's a 3rd. . then a 5th then a bent-up 7th rrelative to the tonic .. ) then maybe this is the way to think of your scales.

Relative to the last note :

Up two tones, down a full tone . .etc
This seems a bit alien to me in that there's little notion of a "Home" note (ie the tonic) but if that's how you hear tunes then maybe this is a way forward.

To me, thinking in terms of an interval from the tonic is most intuitive as it means remembering only one instance of a scale, and is transferrable up and down the neck. The relevant scale becomes a "shape" of notes on the neck whcih makes life easier.

It also works on other instruments, although if you move to a piano you start to see that pianists may think in terms of absolutes, or have to remeber different 'shapes' for each key as the black/white notes translate to different patterns depending on the key.