I am afraid this is due to a misunderstanding with of what Dirac meant. He does not write "for any $y\in \mathbb{C}$" but he refers to a "c-number". The c stands for "classical" as opposed to quantum, and what Dirac means is that $y$ is a real number and not a Hermitian operator. Dirac never considered the delta function of a complex argument, only of real numbers.
When working with a complex number $z$, you can introduce the product of the delta function of the real and imaginary parts of $z$, and if you wish you can call that $\delta(z)\equiv \delta(\Re z)\delta(\Im z)$. So ultimately the fundamental object remains the delta function of a real number.
Concerning Mathematica: I am not able to reproduce your finding that Mathematica would return the delta function of a complex number. if I input
Integrate[Exp[I*lambda*x]*Exp[c*x]/Sqrt[2*Pi],{x,-Infinity,Infinity}]
into the online Mathematica interface at Wolfram Alpha it returns "integral does not converge", which seems to me to be the only sensible answer (without further information on $\lambda$ and $c$).
