What is imaginary infinity, $i\lim\limits_{x \to \infty} x = i\infty$?

In the reals all non-zero numbers have a parity. Either they are positive or they are negative. $\lim_{x\rightarrow \infty}|ax| =\infty$ (if $a \ne 0$) because the magnitude of $ax$ gets infinitely large.

If $a > 0$ then $\lim_{x\rightarrow \infty}ax = \infty$ because the magnitude of $ax$ becomes infinite and the parity of all $ax$ is positive.

If $-a < 0$ then $\lim_{x\rightarrow \infty}-ax = -\infty$. What's the difference between $-\infty$ and $\infty$? Neither of them are actual numbers. Well, again, the magnitude of $-ax$ becomes infinite. But the parity of all $-ax$ is negative so instead of increasing infinitely "in the positive direction", $-ax$ increase in the "negative direction". So $-\infty $ indicates infinite magnitude- negative parity.

Non-zero Complex numbers do not have a single bidirectional parity. A complex number has two components, a real one and an imaginary on and thus are two-dimensional and instead of having a single positive/negative parity, they have a directional angle called an argument. These angles can be in any of an infinite number of "directions" from $0^{\circ}$ to $360^{\circ}$. The number positive $1$ has an argument of $0^{\circ}$. Then number $-1$ has an argument of $180^{\circ}$. The number $\frac 12 + i \frac {\sqrt{3}}2$ has an argument of $30^{\circ}$.

And $i$ has argument $90^{\circ}$.

So what happens to $ix$ as $x \rightarrow \infty$? Well just line $ax$ and $-ax$ its magnitude increases to infinity. So $\lim_{x\infty} |ix| = \infty$. But what is the argument of all the $ix$? They all have an argument of $90^{\circ}$. But $\infty$ means infinite magnitude-positive parity. And $-\infty$ means infinite magnitude- negative parity.

Neither of those apply for $\lim_{x\infty} ix$ which will have infinite magnitude - $90^{\circ}$ argument. How can we indicate that?

Well.... if $-\infty$ means negative parity and $+\infty$ means positive parity, shouldn't $i\infty$ mean $90^{\circ}$ argument?

Briefly, Wolfram|Alpha preserves the $i$ because it's giving you a "direction" for the infinity. Just like $-7\displaystyle{\lim_{x\to\infty}}x=-\infty$ (the direction being "leftwards" on the real line/in the complex plane), $i\displaystyle{\lim_{x\to\infty}}x=i\infty$ (the direction being "upwards" in the complex plane).

This can be used to do many things. To steal an example from the documentation for DirectedInfinity, you can ask Wolfram|Alpha to find an approximation of $\arcsin$ for numbers of the form $\varepsilon+yi$ where $|\varepsilon|$ is very small and $y$ is a very large positive number. Wolfram|Alpha link

This is contrasted with Wolfram|Alpha's confusingly-named "complex infinity", where there is no particular direction. See ComplexInfinity for documentation.

Real numbers and imaginary numbers are different things. ∞ is different from ∞i, just like infinity oranges and infinity bottles of juice are different things. There are operations that will convert one to the other, but that is beside the point.

I think a good way to see this is to see where the reals and imaginaries lie on the Argand Diagram (complex numbers and nonintegers omitted for brevity / clarity.)

                              ∞i

                               .

                               .

                              3i 

                              2i

                               i

   -∞    .    .   -3  -2  -1   0   1   2   3   .   .   ∞ 

                              -i

                             -2i

                             -3i

                               .

                               .

                             -∞i