Antenna, phase response

Lets say I needed an antenna, 5.7-5.9 GHz, that had the most well behaved phase change vs Azimuth and Elevation. In other words, if I measured the wireless transmission phase between it and a receiving antenna a constant radius away in any direction, I would want it to look like the same wireless phase lag. What type of antenna would I use? Could I use a ground plane, or would it have to be some sort of dipole?

I would consider a dipole first. A monopole would have pattern disturbances near the ground plane (and below) that would result in distortions of the phase front.

You might wnat to consider an end fed variation on the dipole to keep the pattern less disturbed by the feed. I wonder if a biconic might be a better choice.

5 to 6 GHz is relatively easy to work with, particularly if you can get to a good machine shop.