What are the effects of adding power to a 45 degree banked aircraft? In a recent question it was proposed that adding excess power to an aircraft in a level 45 degree banked turn would cause it to pitch up into a rolling climb.

This case was compared with level flight using a positive static stability (CG forward-tail down force) model.

In level flight, adding excess power will initially cause an increase in speed, followed by increase in tail downforce,  followed by a pitch up, followed by speed stabilization and climb.
Pitch controls speed, power controls altitude.

Roll 45 degrees and do a level coordinated turn around a point. Add in power.
What I am seeing is a turn around a barber pole with no change in roll, an ascending spiral.

If we added more pitch,  leaving the throttle, rudder and ailerons where they were, a slower upward spiral with steeper barber stripes?

The plane does not seem to roll, the wing is always pointed to the same pole.
Is there any rolling motion WRT aircraft performing these manuvers?
The key is that you qualified the turn as coordinated.  If the turn continues coordinated after the application of more power, then the aircraft will climb due to increased energy being converted to altitude.  Unless you decide to "lower the nose" and turn at a faster airspeed.