Mathematica would leave $\infty+i*\infty$ there, no more simplification.
Before I ask the question, there are number of assumptions I have. If any of them is not right, please point it out. Thank you!
I assume what Mathematica means when displaying $i*\infty$ is $\infty$ in imaginary positive direction. Because $(1+i)*\infty$ would be simplified into $\frac{1+i}{\sqrt{2}}\infty$. The first part is simplified into unit length. Besides, there is only one infinity in Riemann sphere.
If assumption 1 is right: I also think $\infty+i*\infty$ evaluate into $\infty$ which can have any direction between real positive direction and imaginary positive direction inclusive, and that's why Mathematica leave the expression alone.
If these two assumptions are both right, here comes my question:
If the angle between directions of lhs and rhs infinity in an addition is $180~^{\circ}$ (or $2\pi$), the result is indeterminate, or undefined. Except this, is $\infty$ with a range of direction complex infinity ($\tilde{\infty}$)?
Thank you again!
Although this question is in some sense "not about Mathematica", it is confusing for most readers unfamiliar with Mathematica, because the definition of "Complex Infinity" used in Mathematica is not used in any other contexts/texts:
Certainly, if $(a_n)$ and $(b_n)$ are real sequences tending to $\infty$ at (essentially) the same rate, then $\lim \arg (a_n+b_ni)=\pi/4$, a known quantity. Alternatively, if $(a_n)$ and $(b_n)$ are real sequences tending to $\infty$ at unspecified rates, then even though $\lim \arg (a_n+b_ni)$ may not exist, the arguments (and potential limit value) of $a_n+b_ni$ stay in $[0,\pi/2]$.
If either case captures what you meant by $\infty+i\infty$, then the answer may be "it's not Complex Infinity because you know at least some bounds on the arguments; the limiting behavior of the argument is not completely unknown or undefined."
That may or may not explain why Mathematica does not turn $\infty+i\infty$ into $\widetilde{\infty}$. That would be a question about Mathematica.