Any tips/comments regarding learning German from scratch (currently started B1 level) to work as anaesthesiologist in public hospitals in Basel or Rheinfelden?
I have several tips. Aim for C1 level, even though B2 level is legal minimum for licensing. Most hospitals require C1, and since you are limited to one tiny geographical area, you will want to maximize your chances by acquiring excellent language skills, which will almost certainly be your employer's top priority.
Expose yourself do regional dialect as soon and as much as possible - even though it will not be a prerequisite for employment, not understanding it will make your life a daily struggle.
I hope your medical school diploma was issued by an EU/EEA country, which in combination with satisfactory language skills will lead to direct recognition of your foreign diploma. Otherwise, you will face an additional uphill battle.
For answers to any questions about the procedures for recognition, I highly recommend working through the huge and detailed thread started by drremobond007. He is in Indian doctor who moved to Switzerland, and has very kindly documented the process for both EU and non-EU citizens, with qualifications that are from the EU or non-EU.