Consider a (real) projective plane which is topologically
locally flatly embedded in

*S*^{4}. It is known that it
always admits a 2-disk bundle neighborhood, whose boundary
is homeomorphic to the quaternion space

*Q*,
the total space of the nonorientable

*S*^{1}-bundle over

with Euler number

,
with fundamental group
isomorphic to the quaternion group of order eight.
Conversely let

be an arbitrary locally flat topological embedding.
Then we show that the closure of
each connected component of

*S*^{4} -

*f*(

*Q*) is always
homeomorphic to the exterior of a topologically
locally flatly embedded projective plane in

*S*^{4}. We also show that, for a large class of
embedded projective planes in

*S*^{4}, a pair of
exteriors of such embedded projective planes is always
realized as the closures of the connected components of

*S*^{4} -

*f*(

*Q*)for some locally flat topological embedding

.