I am aware of the generalized Bernoulli numbers, but these are not what I'm looking for. I was wondering if there exists such a thing as fractional, real or even complex Bernoulli numbers ( $B_z$ for $z \in \mathbb{C}$).
My motivation comes from the Ramanujan Summation, as the Bernoulli numbers are involved in it. I was hoping that, if the Bernoulli numbers could be extendend, so could perhaps the Ramanujan Summation, allowing it to assign a sum to a wider class of divergent series.
Ramanujan himself gave a definition of Bernoulli numbers for a complex index - see Ramanujan's Notebooks - Part 1 by Bruce C. Berndt, Chapter 5 equation (25.1) with further results in Chapter 7.
Added by J. M.:
Ramanujan's definition of the Bernoulli numbers, as given in Berndt's book, is
$$B_s^\ast=\frac{2\Gamma(s+1)}{(2\pi)^s}\zeta(s)$$