Depends on the problem itself, but if you want to be able to handle any problem it's Complex. If you only want to determine the problem type but not solve it, you can get away with Complicated in most but not all cases.
Generally speaking, an investigation process can determine if a problem is of the same level as the investigation itself. This is often used in customer support, where the first investigation is Obvious and based on scripted questions and work flows. If the problem is obvious, the solution is already known and will eventually be found in the script.
If the problem is not obvious, then this type of investigation will only reveal that the problem could not be solved this way. So you send the problem along to a Complicated investigation process. This would usually require a person experienced with the field to see if the problem is similar (but not exactly the same) as other problems they have encountered in the past.
If the problem is not complicated, then you can assume that it is complex. Just make sure that the problem is actually well defined. If you have a Disordered problem where there is a big disconnect between the actual problem and the way the problem is communicated and understood, you may have a situation where you later find out that the problem was obvious all along but misunderstood. This will be found out once you try to solve it as a complex problem, which is generally a waste of resources.
Actually solving this problem would require experimentation and dialogue with the customer/client in order to gain knowledge of the issue and to try out different ideas for solving it. However, before doing this there is still the need to carry out a Complex investigation to try to get a feeling for if the problem is worth putting effort into, based on the limited information available.
This flow, or some simplified version of it, can be handled by only having a Complex investigation structure in place, but that it usually isn't cost effective because of the high need for a variety of experienced (and probably highly paid) people.
As a side note, if you do happen to encounter a problem that is actually Chaotic for you, this is usually a sign that there are issues with your organisation and your way of conducting investigations. This is more of a responsibility of your company leadership, so that processes and structure can be established first before any work is done. At this point, it should be possible to convert these types of issues into another easier type, by just making someone empowered to actually perform any investigation at all.
