The first hyperoperation $H_1(a, b) = a + b$ is addition, and is defined over the complex numbers. The second hyperoperation $H_2(a, b) = ab$ is multiplication, and is also defined over the complex numbers. The third hyperoperation $H_3(a, b) = a^b$ is exponentiation, and can be analytically extended from the natural numbers to the complex numbers for most complex bases $b$ and exponents $p$. The same is true for tetration $H_4(a, b) = \ ^ab$ and higher-rank hyperoperations.
My quesiton is then can the rank variable be extended to the rational/complex numbers: do there exist fractional-ranked and complex-ranked hyperoperations $H_n(a, b)$, where $n \in \mathbb{Q}$ or $n \in \mathbb{C}$?
