All the definitions of limit superior and limit inferior I have seen (even in the books about complex analysis) define them for a real sequence only.
What could stop us from defining it as follow for a complex sequence?
$$\limsup\limits_{n\to\infty} z_n := \lim_{n\to\infty}\Big(\sup\{|z_k|:k \geq n\}\Big)$$
$$\liminf\limits_{n\to\infty} z_n := \lim_{n\to\infty}\Big(\inf\{|z_k|:k \geq n\}\Big)$$
Best Answer
This feels like a poor generalization of these properties. One of the most valuable properties of the lim sup and lim inf is: 1) they always exist and 2) when they're equal, the sequence is convergent and converges to the limsup and liminf.
Worse, they don't even generalize, in the sense that we'd have for the constant real sequence $a_n = -1$ that $\limsup a_n = 1$ and $\liminf a_n = 1$, which is clearly not the limit of $a_n$.