I took a second glimpse into the paper. Because I am not a mathematician I cannot judge the paper on the mathematical level.

However, as a computer scientist, I can say that it is really strange if a *meaning is not invariant under syntax*. Specifically, the author defines a subobject classifier to be an object $\Omega$ together with a morphism $\mathit{true} \colon 1 \rightarrow \Omega$, such that for every mono $m \colon S \rightarrow A$ there exists a unique morphism $\chi_m \colon S \rightarrow \Omega$ such that the diagram:
$$\require{AMScd}
\begin{CD}
A @>{m}>> S\\
@V{!}VV @VV{\chi_m}V \\
1 @>{\mathit{true}}>> \Omega
\end{CD}$$
is a pullback and claims that a topos is a category with finite limits, finite colimits, exponents and subobject classifier. Then, in section 3, he introduces the concept of a "complement topos" as a category with finite limits, finite colimits, exponents and a "complement-classifier". Where, according to the text, a "componnet-classifier" is an object $\Omega$ together with a morphism $\mathit{false} \colon 1 \rightarrow \Omega$, such that for every mono $m \colon S \rightarrow A$ there exists a unique morphism $\overline{\chi_m} \colon S \rightarrow \Omega$ such that the diagram:
$$\require{AMScd}
\begin{CD}
A @>{m}>> S\\
@V{!}VV @VV{\overline{\chi_m}}V \\
1 @>{\mathit{false}}>> \Omega
\end{CD}$$
is a pullback (yes, I obtained this paragraph by copy-paste-rename).

On the other hand, from Theorem 1 from the paper one may infer that the author uses some inconsistent variant of the meta-logic (i.e. every statement is true in the logic) --- which is quite reasonable --- taking into consideration that his work, as he admits, is built on the foundations of "inconsistent mathematics".

I also consulted the paper "Bi-Heyting algebras, toposes and modalities" by Reyes and Zolfaghari. Here is their characterisation of toposes with co-Heyting internal (thus, bi-Heyting) logic:

Proposition 3.2: A topos $\mathcal{E}$ is bi-Heyting iff there is a Boolean topos $\mathcal{B}$ and a surjective geometric morphism $\Gamma \colon \mathcal{E} \rightarrow \mathcal{B}$ such that the canonical $\delta \colon \Omega_\mathcal{E} \rightarrow \Omega_\mathcal{B}$ has a left lax adjoint.

BTW, showing that presheaf toposes are bi-Heyting without using the above proposition is a good exercise.

ps. My surname is Przybylek, not "Przyblek".

closedsets of a space, but I haven't checked. – David Roberts Jun 22 '13 at 0:42