About the job prospects and skills learned during physics/math PHD I am nearing the end of my degree (which is equivalent to a double major B.Sc in mathematics and physics) and the time has come to specialize in a field. The chliche is almost obvious, I am having a hard time choosing between studying:


Fundamental physics; high-energy theory. Due to the lack of knowledge I cant be more specific.
Mathematics; Mostly of the analysis sort (in the intersection of geometry, functional analysis, probability). Perhaps with a more applicable touch.


I have not considered anything else. CS / engineering do sound nice and exciting, but I know too little about either to determine if I may enjoy them. 

To make sure this isn't the standard "math vs physics" topic; I will attempt to detail as much as I can, and pose answerable questions.

After lengthy contemplation, I am still unable to realize towards where each academic degree would lead me. Academia is an option, but, adopting a more realistic point of view, the number of positions is small and I cannot foresee whether I will still like academia. This pushes me to explore other possibilities. Fortunately (or unfortunately) I have enjoyed both topics very much, and studied them with success.

To avoid unnecessary repetitions, in the rest of my post, physics would mean high-energy physics (phenomenology or theory). Math would mean geometry, functional analysis, probability.


What skills, applicable outside of physics/math, does one acquire in physics/math PhD?
What can a physics/math student do during their PhD to increase their job applicability? I refer to this What skills can I get during a PhD to be competitive on the job market against non-PhD holders? but would like to note that the former is broader, as the possible PhD there is not mentioned.
When physics/math PhDs leaves academia, they would rebrand  themselves, for example, as engineers or data scientists. How difficult such a transition is? Asking friends, professors and searching through the internet shows they get to high-tech, big data, financial trading. But more specifically, how could a theoretician/mathematician develop the skills to become, for instance, chief engineer (maybe simply engineer)?
Are these degrees appreciated outside of academia? For comparison, let's say CS and engineering are appreciated. How bad does math/physics compare to them? I apologise this question is not very subjective, but I get the feeling that although people who go through these programs are capable of dealing with such abstract concepts - they are disregarded outside of academia.
Studying any of these, can you get involved in developing algorithms fairly easily? What about AI? This sound like things I would like to do if academia doesn't work out for me.


I find this: MS degree after a PhD in physics demoralising, but would like to understand what I should expect. Expiriences like that probably answer some of my questions, and are very welcome.
As a preamble, I don't quite understand why you suggest that fundamental physics is connected to high energy physics since the bulk of recent foundational experiments have been done with lasers and atoms, but to each his/her own.

The large majority of physicists work in industry and do quite well, as you can see from the salary chart below (source).  



Employment tends to be related to solid state technology, information technology or optical technology (the areas that produce the largest number of graduates) but even graduates in less industry-obvious specialities like high energy physics find industry jobs. 

The running joke is that a physics and an engineering graduates are sitting next to each other at a graduation ceremony.  The engineer is happy to get a degree, but the physicist replies that he (because there aren't that many female graduates) not only got a degree but also an education.  

PhDs in physics have a capacity to reason abstractly and comfortably deal with the abstract.  As a PhD candidate, you will have to learn a good deal of programming on various operating systems.  You will be faced with challenges of data integration (or data fusion) and data interpretation, understanding filtering of data etc (especially true if you do high-energy phenomenology); these are but examples of a number of techniques that have wide direct industrial applications, irrespective of the field you specialize in. 

Transitioning to industry is not necessarily hard, but you will not get an industry job because your thesis was on 10-dimensional supersymmetry: some flexibility is required.