Since the 2010's, entropy measures have been extended from the 1D to the 2D case to deal with images and are forming potent bidimensional irregularity measures. In our work, we study a new 2D entropy measure - the so-called bidimensional fuzzy entropy (FuzEn) - that out-performs existing bidimensional entropy measures. We first assess its sensitivity to parameters, then analyze its behavior upon rotation and translation, and finally show its multiscale application in the biomedical field (dermoscopic images). To validate the output of the newly introduced FuzEn and its multiscale extension, a set of synthetic images based on several concepts in image processing (including power of noise and degree of randomness) and texture datasets are used. The results for synthetic images illustrate that FuzEn has low sensitivity to the chosen parameters and it is rotation and translation invariant. Moreover, it outperforms the already existing bidimensional entropy measures. Finally, we evaluate dermoscopic melanoma (malignant lesion) and melanocytic nevi (benign lesion) images and the results are found to be interesting for a potential diagnostic application.