The closest thing I wound in Diophantus is problem IV(24) which is solving the system
$$X_1+X_2=a,\quad X_1X_2=Y^3-Y.$$
Diophantus sets $X_1=x$ and eliminates $X_2$ obtaining
$$x(a-x)=Y^3-Y.$$
This is your equation if we set $a=i$ and $y=ix$. 
This seems to be the first elliptic curve encountered in the book of Diophantus; before that he only considers rational curves and surfaces.

This little research is based on a Russian translation of Diophantus with
comprehensive comments by I. G. Bashmakova, published in Moscow in 1974.)