If you are already in Doha and you want to go to anotehr company, if the company object to you 'leaving' they will not transfer your visa. So you can not work for the new company.

However, a new company who may wish to employ you will ask if your visa trasferable ie: your old employer will allow you to leave and work with another company (normally the old company will issueing letter of no objection or something like that).

This is my understanding, maybe somebody out there can give you more detail.

E4 maybe you can help him :-)