I know that this is more a soft question and that the answer depends on the answerer but I think that for every discipline it is not only essential to discuss the discipline itself but also its methods and the process of its creation.
It might be very fruitful to discover how other people think about Mathematics before it is put into a readable form. Furthermore, when not being subject to certain conventions and limitations of writing, the freedom of the thoughts might reveal some beauty of mathematical ideas that could not be perceived when only discussing the result of these thoughts.
Thus, I would like to know what you imagine when you do Mathematics.
For example, do you move on the number line when calculating in your head or do you follow the graph of a function when integrating or do you visualise open and closed sets when doing a proof in topology? And how do you visualise that? Do open and closed sets have different appearence or are they only different by a certain feeling that you associate to them when thinking about it? Do you think Mathematics in color? For example when thinking about the golden ratio or about a plane that is crossed by a line? Or about higher dimensional space - does it still look like a 3D space but with different rules? Do you hear Mathematics? Are there different sounds that you can associate with fractals for example? Or with infinite serieses? These are just ideas to inspire an answer, if you imagine things completely different from these, I am also glad to read about it!
What I am also interested in is the question whether there are people who think more in terms of words and definitions without referring to any visualisations, or if there are people who can think about the abstract without needing the concrete - for example for me personally it is hard to think about any proof in a mathematical field before not having at least one example. And the more examples I have, the easier it gets for me to interpolate between them, to see solutions before all details are worked out.
Personally, I am often in a realm between an exact visualisation, facts I remember and logical implications. For example, I do visualise how numbers are seperated as strokes on a line when calculating with them but then I switch fluently to a picture where I only see single numbers and apply certain rules of calculation that I know. Or I associate an equation in a precise way with a physical process but when calculating with it I only see the equation itself. It is often a mixture where visualisations act as corner stones of orientation in a logical web.
And when a conclusion fits together with another thought it feels a bit like the relief after drinking something very fresh. (I hope there are not too many psychologists who will read this.. ;-)
So please feel free to add what you imagine and maybe a bit of the beauty of what you perceive in a unique way. Thanks to all who can share my smile about that.