New Shepard is described as using liquid oxygen and liquid hydrogen for its propellants. When hydrogen and oxygen burn, as in the Space Shuttle main engines, there is a short conical blue flame, beyond which the exhaust is clear (until the water vapor condenses). The New Shepard exhaust appears to have a long, distinctly red/orange/yellow flame, which would suggest that there is something else burning besides hydrogen and oxygen. What's going on?
The first few feet of exhaust from the New Shepard is clear, and the reddish plume fades with altitude, so we can rule out an ablative liner like the Delta IV's RS-68 engine uses -- that would produce a more consistent yellow plume.
Hydrogen's emission spectrum includes lines in ultraviolet, violet, blue, green, and red. Very hot hydrogen combustion, like that occurring in the engine's combustion chamber, gives more of the violet and blue emission, but the exhaust is fuel-rich, and cooler than the combustion chamber due to expansion in the nozzle, so when the exhaust hydrogen combusts with atmospheric oxygen at lower temperatures, you see more red.
The space shuttle's RS-25 main engines also produce a little pink light in the exhaust plume, but it's faint compared to the very bright plume of the solid rocket boosters, so not so noticeable in photos. The exact exhaust chemistry may be different as well; I don't know how the propellant mixture ratios compare.
The DC-X also used regeneratively cooled, hydrogen-oxygen RL10 engines with no solid rocket boosters, and shows similar pinkish-red plumes: