Well, all along a "Profile" problem seemed likely. My printer was an early suspect, but good nozzle checks, new carts and using HP's diagnosis app, it didn't seem that printer problems were the culprit. So I thought, what about letting the printer take care of color output?
In the various menus I set the printing to let printer manage color. There were two options -"Let printer/driver manage color" and then the ability to choose between sRGB and AdobeRGB.
I put this in the wrong post. It's my final summary for now.
I went for no application color management. The setup is "Let printer/driver manage color.
I chose sRGB first and there was no improvement. Not expecting a better print I then chose AdobeRGB. An excellent print was the result of that.
In summary, printer manages color using Adobe RGB is good. All else, using a profile done for me and multple HP profiles and printer manages color using sRGB all gave the same amount (apparently) of magenta excess.
I'm not a newby at this, I've done lots of printing on Canon and HP 24" printers using Photoshop initially in 2005, and Qimage for 9 years. I know how to print unmanaged samples for profile creation etc.
I'm currently trying an earlier Qimage on a Windows 8.1 machine to see what happens. I'll report on that.