You could say "I'm wanting a new challenge". But in this case, you've got to be perfectly aware you're breaking a commonly accepted usage. If you're doing it on purpose, because you intend to be more specific, because you want to convey some very special connotation, then you're making the language richer.
of course, as a non native, this is something I would do with extreme caution, if at all.
Conversely, if you use it as an interchangeable substitute for "I want a new challenge", then you're making the language poorer.
Only my opinion (which can be extended to the use of other stative verbs in the continuous and, more generally, to all "unusual" wordings).