  We study a Bose-Einstein condensate in a PT-symmetric double-well potential
where particles are coherently injected in one well and removed from the other
well. In mean-field approximation the condensate is described by the
Gross-Pitaevskii equation thus falling into the category of nonlinear
non-Hermitian quantum systems. After extending the concept of PT symmetry to
such systems, we apply an analytic continuation to the Gross-Pitaevskii
equation from complex to bicomplex numbers and show a thorough numerical
investigation of the four-dimensional bicomplex eigenvalue spectrum. The
continuation introduces additional symmetries to the system which are confirmed
by the numerical calculations and furthermore allows us to analyze the
bifurcation scenarios and exceptional points of the system. We present a linear
matrix model and show the excellent agreement with our numerical results. The
matrix model includes both exceptional points found in the double-well
potential, namely an EP2 at the tangent bifurcation and an EP3 at the pitchfork
bifurcation. When the two bifurcation points coincide the matrix model
possesses four degenerate eigenvectors. Close to that point we observe the
characteristic features of four interacting modes in both the matrix model and
the numerical calculations, which provides clear evidence for the existence of
an EP4.
