  Given a complete K\"ahler manifold $(X,\,\omega)$ with finite second Betti
number, a smooth complex hypersurface $Y\subset X$ and a smooth real $d$-closed
$(1,\,1)$-form $\alpha$ on $X$ with arbitrary, possibly non-rational, De Rham
cohomology class $\{\alpha\}$ satisfying a certain assumption, we obtain
extensions to $X$, with control of their $L^2$-norms, of smooth sections of the
canonical bundle of $Y$ twisted by the restriction to $Y$ of any $C^\infty$
complex line bundle $L_k$ in a sequence of asymptotically holomorphic line
bundles whose first Chern classes approximate the positive integer multiples
$k\{\alpha\}$ of the original class. Besides a known non-integrable
$(0,\,1)$-connection $\bar\partial_k$ on $L_k$, the proof uses two twisted
Laplace-type elliptic differential operators that are introduced and
investigated, leading to Bochner-Kodaira-Nakano-type (in-)equalities, a
spectral gap result and an a priori $L^2$-estimate. The main difference from
the classical Ohsawa-Takegoshi extension theorem is that the objects need not
be holomorphic, but only asymptotically holomorphic as $k\to\infty$. The
possibility that $\bar\partial_k$ does not square to $0$ accounts for its lack
of commutation with the Laplacian $\Delta''_k$ it induces. We hope this study
is a possible first step in a future attack on Siu's conjecture predicting the
invariance of the plurigenera in K\"ahler families of compact complex
manifolds.
