I have carried a compass for all of my touring life - first my good silva work compass and latterly an electronic one in my Suunto watch.
However, whilst I am very profficient in the use of a compass - I was taught at an early age to use one by my father who was a professional navigator and I often use one at work - I have never used one for navigation while on the bike. If I am hill walking the use of a compass - even if I have a GPS - is a given. The compass does come into its own in the evening when I am setting up my tent - so that -hopefully- I will get some early morning sun.
I think because I carry my map on a holder on the handlebars in front of me in sight all the time it enables me to have a good mental picture of where I am and are going - tend to be comparing the map all the time with features I can see on the map and the ground - other roads, rivers, the coast, railways etc - plus I am always subconsiously monitoring where the sun is (if I am so lucky) and the wind direction.
Much as I hate to agree with Cunobelin - I did experiment with a little bell with a compass (a gift) on the handle bars but after a while it soon became apparent that it could be as much as 45° difference from the true - my own mental view of which way up the world was was generally more accurate.
Nonetheless, if you think using a compass would help you then by all means take one - just be wary about how you use it