The medcouple has no units. Certain complications aside, which don't produce an exception to that statement, it is a ratio of quantities on the original scale, so the units cancel. It could hardly be a (useful) measure of skewness otherwise.
Personally I often value sensitivity in measures of skewness (and kurtosis, which otherwise we will leave on one side here), as they alert me to what may be a problem, or even a feature: high skewness may be one signal that I would be better off working on some transformed scale.
For those in search of alternatives to moment-based skewness:
It's always worth remembering that (mean $−$ median) / SD is bounded in $[−1,1]$ and is easy to explain and calculate. It's not particularly robust, but as a bounded measure it is less explosive than the moment-based measure.
L-moments offer a well-based set of ideas. The Wikipedia entry is a fair start, with several key references.
The thread Robust analogues of Mean, CV and Skewness also touches on these issues.