Like Antechinus, I noticed this while lying on my side in bed. Regardless of what side I was on, the eye closer to the ground had the reddish tinge. I figured it was blood-pressure related, also. And now I’m shocked to see that so many others have noticed the same thing.
Perhaps the two eyeballs, having slightly different prescriptions, are straining themselves to see correctly in slightly different ways. Maybe this causes uneven pressure in the two eyes, giving one a red shift and the other a blue shift. Or, as was previously suggested, its the doppler shift. 
Are you certain you don’t have a pair of 3-D glasses on?
I notice this when lying on my side, as well.
But here’s a new idea - are you female? Several forms of color blindness are X-linked, meaning the genes for them are on the X chromosome. It’s widely known that color blindness shows up more often in men, because they only have one X chromosome and any defects are not balanced by the other. What’s less well known is that women only have one X chromosome active in any particular cell. So if you carry an X mutation, half of your cells will be defective, and these cells tend to occur in blocks. If, by chance, one of your eyes has a big block of defective color-sensing rods, then I imagine the brain would adjust the other one to even it out. Voila! One eye sees no (or little) red, the other one is correspondingly red-shifted.
If you are male, then this doesn’t apply.
mischievous
I mentioned it to my optometrist when I was a child.
There were test cards with numbers composed of colored dots. I couldn’t see some of them with one eye. He tinted one lens slightly.