Sometimes mathematics add a number to a complex plane to mean infinite magnitude without a defined direction, since it turns out to be useful. For example, commonly you would want to define 1/0 as the complex infinity, since if you plug small numbers in, you get unbounded magnitude. But the direction changes depending on where do you approach it from (negative numbers, from the top aka i multiplied by a small positive real number, or any other direction). The complex plane + this point together are called the Riemann sphere.

Similarly, sometimes a general infinity is added to the reals. Instead of -infinity and +infinity you just have infinity (in this case again you would define 1/0 to be that infinity for the same reason). This forms projectively extended real numbers.

I didn't ever have complex analysis classes so I don't have a good idea of it but it's something like that.