Here I define "strongly open" to be the following:
Given a function $f: (X, \tau_1) \rightarrow (Y, \tau_2)$, $f$ is strongly open iff for any set $A \subseteq X$, $f(A)$ is open.
I came up with an example: Let Y be a set and say $y_1, y_2 \in Y$ and then we define $\tau_2$ := {$B\subseteq Y | y_1 \hspace{1mm} or \hspace{1mm} y_2 \in B$} $\cup$ {$\emptyset$}. $\tau_2$ is closed under arbitrary union and finite intersection so $\tau_2$ is a topology built in $Y$. Now assume $\tau_1$ is a topology built in $X$ and say $A$ is a non-open set in $X$. Define $f (X, \tau_1) \rightarrow (Y, \tau_2)$ such that $f(A)$ = {$y_1$} and $f(A^{c})$ = {$y_2$}. In this case, $f$ is strongly open and in fact $A$ can be substituted by any other subsets.
Also, one can replace $y_1, y_2$ with any finite subset $S$ in $Y$, build up $\tau_2$ and $f$ in the same way as long as we partition $X$ into pieces with the same size as $S$'s. Obviously following the same idea, we can fix any subset $B$ in $Y$, define $\tau_2$ to the set of sets with non-empty intersection with $B$ along with $\emptyset$, define a partition in $X$, which is bijective to that fixed subset in $Y$ and then define $f$ accordingly. In these cases, $f$ is strongly open.
I would love to see a more complicated example than mine (ideally a real/complex valued function or you can choose to prove that such a function will not exist). If you think your example is smarter than mine, I would be excited to see it :)
The function $f$ is strongly open if and only if every subset of $f(X)$ is open in $Y$. In other words, $f(X)$ must be open, and the induced topology on $f(X)$ must be the discrete topology. This essentially means that all examples will be "boring".
In particular, for every $x \in X$, the singleton $\{f(x)\}$ must be open in $Y$. This proves that there are no real or complex valued functions which are strongly open, because no singleton in $\mathbb{R}$ or $\mathbb{C}$ is an open set. (I guess there is one trivial exception: if $X = \emptyset$, then the empty function is strongly open.)