Computer, mostly hardware (so heavy on electronics rather than programming)

Oh yes no doubt j is very useful - but that's what I meant by it being part of the Toolbox. To visualise it as an entity is something else though, and I know that you're not really supposed to do that, especially when it was 'invented' to be useful in the first place.

Perhaps with more experience and perseverance, I might have reached an understanding like yours. I do find myself regretting for giving up on engineering so soon, but hey, I can still take my time to understand what I couldn't before. Thanks for the links :)

You can and should visualize imaginary and complex numbers. Whoever told you otherwise was wrong. See my other post. Plot a complex number (say 2+4i) and then try two operations on it: multiply by 2+2i (this will rotate the point to the left and scale it); and try adding a complex number (say 1-3i) which will have the effect of translating the original point.

Developing an intuition about what rotation a particular complex number will give you is more complex, admittedly. But this is enough to visualize the effects, if not precisely mentally predict the result.

I was thinking about your post yesterday, thanks for it :) Sorry, I didn't clarify with what I meant by "visualising." When you think of an abstract entity, like a foreign word for example, you can sort of grasp what it means when you have it translated. But you can't do that with the imaginary number. You can only get a 'feel' of what it is when it's set in context, like the expressions you gave, and even then the application is still in a math world (geometric (?) space.)

In hindsight, I think that I approached it wrong, and it was far too simplistic anyway. I've been reading popular maths books lately and it's made me realise that there are more ways than just visualisation to understand something. Ironically (a happy one!) this is making me excited about maths for the first time in my life.