  This paper investigates the connection between blow-up solutions of scalar
reaction-diffusion equations, in particular of $u_t = u_{xx} + u^2, $ and its
counterpart - eternally existing solutions like heteroclinic orbits - by
complex time. We prove that heteroclinic orbits in one-dimensional unstable
manifolds are accompanied by blow-up solutions. Furthermore we show, that we
can continue blow-up solutions into a slit complex time and eventually back to
the real axis. The solution picks up an imaginary factor after continuation
which is related to the eigenvalue relations of the linearizations at the
source and the sink of the heteroclinic orbit.
