As you have seen, you can define the behavior of existing operators, like + or * to operate as you would like them to behave for objects of your class. I use that capability heavily in several of my toolboxes.
But no, you cannot define a completely new character and have MATLAB recognize that as an operator. So you cannot define the operator $ or ° to do something special for your class. This may change one day - one can never rule out language design changes.
