Can you show an example? I can't imagine what signal it would be calculating a doppler shift on, that could be more accurate than the positional data it computes. Most of the satellite signals -- certainly the strongest -- will be high overhead. Thus the signal path is essentially perpendicular to your direction of travel, and doppler shift wouldn't be significant.
Seems to me it won't be long til GPS units simply include a six-axis accelerometer, since acceleration can be directly measured without need of GPS and since such tech is cheap and already in both smartphones and game systems like the PS3 and Wii.
This will allow for far more precise tracking of your speed at any given moment.
Except that integrating any real world signal tends to introduce drift due to noise in the input signal. If you have a smartphone, try writing an app that does exactly what you propose; you'll soon find that it's impossible to get the "speed" to zero out. (Either that or you have to filter the data, which will cause other fun errors like your speed drifting toward zero.)
I thought about this as a complication, but my assumption was that this would be a solvable technical problem. Is it really not feasible to add an accelerometer to the GPS data (which does include altitude), and end up with at least somewhat more precise data?
You can feed the accelerometer data into the Kalman filter as well. However, since it's much more noisy than the GPS data, it won't give you any real benefit.
Where people would like this is to do dead reckoning when a GPS position isn't available (e.g. in tunnels). However, it turns out that it is completely useless for that, for the exact reasons the GP describes.
Actually, an accelerometer is far, far less precise because something like going up a slight incline results in a pretty significant change in the horizontal acceleration but a comparatively small change in the vertical acceleration (which is easily lost in the noise). The rotation caused by starting to go up a hill is also so tiny that it is completely lost in the noise.