I have a few questions about supplementary insurance,

If you have your supplementary insurance and basic insurance with different companies, how on earth does the billing work? I would assume bits would be paid by compulsory bits by supplementary.... This would be a nightmare?? You only get one bill, so how can you send it off to both insurance companies?

Also what about the franchise? Would you have a franchise for each company?

Most conditions are handled separately by the compulsory and supplementary insurance, so generally it's up to you to send it to the correct one. For example, dental bills, opticians aren't included in compulsory insurance, so you can send them off direct to the supplementary insurance.
At the worst if you send the bill off to the wrong company, you'll get a notification "not covered" or something similar and forward it to the correct company.

But let's say you get treated in a different canton for a condition covered by basic insurance. It costs more than being treated in your home canton. Your basic insurance should cover part and supplementary the difference! How would that work?

I've never had that and my doctor works in a different canton from where I live (I live 250 m from the cantonal border with Geneva), but my understanding is that treatment under basic insurance is valid throughout Switzerland and that it will always be covered by the basic insurance. The only thing I can imagine being different is hospital treatment, but usually hospital costs are directly handled by the insurance.
In any case, I would send the bill to the basic insurance and if they claim that they will only cover up to a certain amount, wait till you get the account from the basic insurance and claim the rest from the supplementary.

Dental is covered by supplementary?!

Basically yes. I'm not a specialist and it is possible that some complicated dental surgical procedures are covered by LAMAL, but the average stuff most of us go to dentists for (descaling, fillings, root channels ....) can be covered at least in part by supplementary.

If you break a tooth in, say, a skiing accident, the accident insurance will pay.

(most) Important things that the compulsory insurance doesn't cover:

Complete hospitalisation in all cantons.
Glasses and contact lenses.
Ecographies (2 max.) and mammographies (for the ladies).
Emergency transport (wich is very expensive), the coverage is only a small part of the costs.
Medicaments off the list (nearly 60% of all)
Alternative medicines.
Dentistry.
Etc.

For those you need to have a complementary insurance, but, don't be fooled, not all the complementary have these coverages. You can have a complementaries only for Dentist bills, another for complete hospitalisation, one with almost everything, it depends on the company.

For exemple, I have one that covers:
Complete hospitalisation in all cantons, glasses, medicaments off the list, alternative medicines, fitness bills (like gym), and then another with coverage in ALL the world (cause you are very limited in other countries with the compulsory insurance). And then the legal protection of course. Switzerland, with all its bureaucracy, papers, bills, traffic fines, it's really usefull.

Well, I work with insurances, so I'm used to explain this kind of things -.- this is a small resume. If someone have any doubts post here or send me a PM.


Answering to the first question: I advise you to do a copy of the bills, and inform the first company that you have complementaries in another company so they send you a document (informing what they've paid or not) for you to send to another company along with the copy of the original bill. Note: usually, complementary doesnt cover franchise nor co-payment of the base insurance.
The franchise it's not the same, you pay the first franchise in the base and the ONLY IF you have another franchise in complementary, you pay for it.

Can an insurance company sell supplemental insurance in CH without selling the non-profit basic health insurance, or must you offer a basic health insurance to sell anything in the supplemental market?

I believe they can sell just supplemental. A few years ago, we had Innova for the basic and in 2011, they shifted (sold, I assume) all their basic customers to Visana under the VivaCare product. Innova now concentrates on the supplemental market, afaik.

Whether there is a complicated corporate structure beneath this, I don't know.