While pursuing a PhD in an experimental field, one usually researches a given topic and conducts experiments that are rather closely related to this topic and one another. I imagine this specialization can be beneficial as one attains expertise in the research field and can work in a timely and focused manner.
However, I can also imagine that this specialization may make changing your area of research more difficult when applying for positions as a post-doc. Therefore, I am wondering about the advantages and disadvantages of researching scientific questions one is interested in, but that are irrelevant to ones PhD thesis on the side.
TLDR: You will probably become a better problem solver if you do it, but there is a chance that it won't count into the formal requirements for graduation!
In general I would say it is not a good idea to research multiple independent topics when you are a PhD candidate, especially if your PhD is supposed to be really narrow. The risk is that you will become distracted from the main focus of your project. Your main focus will then suffer and even if you actually do find results in the topics too far away from the project you are part of those are likely to be judged to not fit the project. Those results may impress other people for example in various industries and maybe academics too, but maybe they will not help you pass the formal requirements for graduation!
( Everything I've written is assuming you value graduation above all else, of course. )