Could you put your desk all the way across under the window and have it as a floating desk with nothing underneath so you can roll your office chair across to stare out the window when talking on the phone for example?

I would love to do that!

The challenge is the window is good size and the base is only a foot or so off the ground, I don't think I should have the desk go across the glass pane.

I think it would be determined by your working preferences. It would be friendlier to face each other across the desks perpendicular to the window unless you find that distracting.