If need to establish a business abroad what will be my tax residency as natual person in that case?
I run an individual business in Switzerland in the field of photography services. I would like to expand the business to a country in the European Union. Should I then set up a business in that country, or will this not be necessary and I can invoice using my current business and then pay income tax in Switzerland?
Since you'll presumably be doing the work in the other countries, you may well have to create some kind of business entity there (could be sole-trader). The rules vary by country. You'll need specific advice from each country.
If you physically in the another country for too long (along with other criteria), you can end up being tax resident in that country and Switzerland. However, any income generated within a country is generally considered to be taxable in that country regardless. Tax residency is about things like if they will tax you on worldwide income.