I understand your confusion and it is really something that bothers many at early stages in complex analysis. I say you forget about it at the moment and focus more on the introductory topics. There is a book by Zill which I consider easy to understand. He introduces transformations in an easy and more intuitive approach than other books.
Most of the books I've read on the introduction to the Complex Analysis, DO NOT talk
about the general picture of it and its usefulness.They immediately dive into the detail and impose the learner to read the stuff before having an clear insight about what is going to happen.
I have got some general questions about the Complex analysis and the world of complex numbers which I haven't been satisfied by the answers given by books yet.
What I have figured out of the subject is, mathematicians defined the square root of minus one to have the power to play with the answers a little bit more, it's kind of like inventing a way to bypass the restrictions.
What I haven't figured out yet is, what we're supposed to do with the extra variable we've got if we only need the manipulation of one of them.For example, consider the $z=x+iy$, if we only need $x$ to solve a particular problem then can we consider the real part of the answer? and don't take care of the manipulation of imaginary part
The serious question is, is it the nature of the complex algebra to solve two problems simultaneously
And the other problem is, I haven't got any intuition about the limit of the complex functions, for example we have intuition of the limit on the graph of the real functions, is it any similar thing for the complex functions?