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As a EU citizen, your wife should normally get a working and residency permit only by proving she has found a job (i.e. Her new employer signs a simple form).

As you are married you should also receive a Residence permit as it is family reunion. If it is the same than in Geneva you can ask for both of your permits on the same form.

If you don't have a job yet, it is possible that they will grant you a residence permit without a working permit. Normally you can switch it to a proper working permit as soon as you find a new job.

So basically, no worries for your moving but it might be a good idea to give a call to Basel administration in order to ask them what would be the process in order for you to be able to work and not only live in Switzerland.

Regarding the processing time, it's usually 6 to 8 weeks in Geneva or Vaud. I can only guess it will be around the same in Basel.

As a EU citizen your spouse should be allowed to work for her new employer during the processing time, as soon as she sends back the paper work to the administation.

As you are a non EU citizen I don't know if it would be the same for you and that's something you might also want to ask to the administration.

PS: In case you're wondering, the differences of treatment between EU citizen and non EU citizen are the result of a special agreement between CH and the EU.