What is the real and imaginary part of complex infinity? I know that complex infinity is a pole on the Riemann-sphere, but what is actually the real and imaginary part of it? Is it $\pm\infty \pm \infty i$? And how is it different from simple non-complex infinity?

Complex infinity is just that: $\infty$. In particular, as far as we can make sense of this at all, we have $\infty=-\infty=i\infty$ (or in fact $\infty=z\infty$ for any finite non-zero $z\in\Bbb C$). It makes no sense to speak of its real or imaginary part or its argument.
In spite of the same symbol being used, the $\infty$ that is added to $\Bbb C$ in its one-point compactification is not directly related to that $\infty$  that is (together with $-\infty$) added to $\Bbb R$ in its two-point compactification. (There is also a one-point compactification of $\Bbb R$, and the $\infty$ used for that may perhaps be identified with the complex infinity if we view the - compactified - real line as a great circle of the Riemann sphere). I'd even say that the uses of the symbol $\infty$ in things like $\sum_{n=1}^\infty a_n$ or $\|f\|_\infty$ are only loosely related to these as well. (And we haven't even started with infinities occurring as cardinalities and being written with a whole different bunch of symbols)
