The set of real numbers is closed under multiplication, but not under exponentiation (Eg. square root of negative numbers). That is, $\exists a, b \in R \mid {a^b} \notin R$. Then we introduced complex numbers and it is closed under exponentiation.
Now that I came across tetration in Wikipedia, is the set of complex numbers closed under tetration? or can tetration between any two complex numbers does not exist in C? That is, $\exists a, b \in C \mid {^ba} \notin C$ is true for any $a,b$?