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?

$$\limsup\limits\_{n\to\infty} z\_n := \lim\_{n\to\infty}\Big(\sup\{|z\_k|:k \geq n\}\Big)$$
$$\liminf\limits\_{n\to\infty} z\_n := \lim\_{n\to\infty}\Big(\inf\{|z\_k|:k \geq n\}\Big)$$