I want to pursue topics from different fields (pure math,CS) as a researcher. Which one to choose for my PhD? My background: I have two master degrees in pure math and applied CS. I am interested in working in both areas, and there are certain topics in each area (extensions of my -completely different- two master theses)  that I am interested in working in, so there does not exist the possibility of combining them in any way: math its 100% proofs and nothing else; computer science it's code a prototype, computing the some accuracy metrics, writing more code.          

However, my working style is to switch between these two field every 9 month, since I quickly get bored, if I only work in one of the fields.

I recently received an offer for a PhD at a top European university (think Cambridge/Oxford/ETH... ). Like any funded place, this one is tied to a single topic from one of those two fields and some compromises have to be made to harmonize my topic choice with the problems my supervisor likes to work on.

1) Do you know of any strategies that I could use to keep switching topics, so to preserve my passion for both fields?
("Utopically", I would get accepted into a second PhD, where I would work on a topic from the other field and then keep rotating every 0.5 - 1 years between them, i.e. as soon as I achieved a milestone in field and start to feel boredom creeping on, go to the other field. But no sane supervisor would accept that, I think. And the program structure would.)

2) Are there ways to convince my supervisor to let maybe 1.5 take year-long time off? That would already help a lot I think. How should I best go about this, without giving a bad first impression ("you haven't even started, and already you think about going away").

3) Or should I just stick with it and try to finish as fast as possible with the PhD and then try to move to the other field for postdoc (supposing I could get a postdoc position in say, math, if I did my PhD in CS, and only published 2 smaller articles in math in the meantime)?

4) If there absolutely no way to switch between topics for my PhD, I'm unsure which field would be the best to go for for my PhD: Pure math (doing proofs) or CS (coding proof-of-concepts in machine learning and related areas)?

I assume many will say things like "this is a PhD! You have to learn how to stick with one topic, because this is life and life can be hard". But  working on multiple problems at once is a tried and tested way to keep the fun in the research (as done perhaps most famously by Perelman: "he liked to work on several problems at once" quote). Alas, if I knew them, I would not post here; I'm really at a loss for what the best step would be to preserve my love for doing research, while not ending up unemployed and keeping the door open for a career in academia.

edit: While the current answer provides very good general feedback, I would still like to know more (very) concrete information about 1)-4).
Generally speaking PhD studies are all about specialization. One dives very deeply into a narrow field. This is especially true of math and can also be true of CS, depending on subfield. 

There can be exceptions and some synthetic work is possible, but it depends a great deal on your advisor and what he or she is comfortable with. If you can find an advisor at the intersection you should be able to pursue a joint approach. 

After the PhD you are much freer to pursue your own goals not constricted by the concerns of your advisor. I would suggest, pretty strongly, that you explore your wishes with a doctoral supervisor and take their advice. If the advice is to pursue the one field for now, it isn't a decision you need to live with for all time. But the advice will likely be good advice. 

Doing a second doctorate will probably be superfluous. You will have deep research experience from the first. If it is in math, much of what you learned will have some application to thinking about some of what is important in CS. 

Doing two doctorates simultaneously seems masochistic to me, especially as you will have two advisors each with different, possibly conflicting, expectations and each putting constraints on your time. 

Think specialize now. Think widening interests for the future. It will better match the expectations of those around you and make life easier. 

But you will also need ways to not get bored, given your description of the problem. It might even be that while focusing deeply on one of the fields you can 'play about' in the other for a short periods just as a stress reducer as needed.