At the Quora website a few mathematicians have been discussing the place of visualisation, or the picturing of concepts and patterns, in maths. (Above is the difference of two squares, “pictured”).
One useful remark: “the best description of what is going on during the process of learning math is that you are ‘getting used to the concept’. It can sound weird, but almost everything is hard at first but when you work with it for a week, a month, a semester, a year… when you hear a word like category, vector space or variety, you don’t try to visualise it, you just know, that it is something familiar to you and you start thinking about it more concretely depending on the particular situation… However, it is always good to have some ‘picture’ of different concepts in your mind…”
Another: “I think that it is always important to try and gain intuition about what you are doing. You should have an internal model that you refer to that should suggest whether a result might be true or not”.
It could be added as a qualification to the first remark that, although it is indeed hard to “visualise” a category, Saunders Mac Lane’s classic Categories for the Working Mathematician says in its preface, “our subject could be described as learning how to live without elements, using arrows instead”. There’s some picturing there, no?
One point missing from the discussion so far is the place in maths of switching between different picturings of the same concept. Important, I think, and not just with complex numbers.