By definition a handrail extension is part of the handrail and the height requirements would apply.
[RB] HANDRAIL. A horizontal or sloping rail intended for grasping by the hand for guidance or support.
But MT,
In the IRC what one calls an extension which are not required, are starting easing's and or returns, there is no limit on their size, and the IRC exempts them from being within the handrail requirements when over landings.
The IRC, under R311.7.8.4 Continuity., as written in exception no. 1, specifically the requirement for continuity, "shall be permitted to be interrupted by...,, at a landing,....."
Thus, the height is interrupted on it's way to returning to the wall.
Though I would agree that if it is not in the IRC, you can revert to the IBC to explain a situation for a requirement, the IRC is explicit here in this case in exempting the height requirement from the nosing to the end of the return by not limiting this area, under exception number 1, the sole reason it exists is to allow for the handrail to be higher or lower.
I don't buy that if something is not required but installed, it has to meet the requirements as if it required.
Bill, if the so called additional handrail is installed completely outside the required heights, 34-38, and there is one at the required height location, then the additional item is technically a handrail at all because none of it is in the right location, which is fine.
However, MT's stand is that the sloped portion of the handrail is within the required zone and that makes the extensions part of the handrail continued, and I agree with him on that point, however he is leaving off the exception specially written into the IRC code to exempt this type of termination, though the first thing that hits me is WTF?, when I see it, a lot, really a lot.....