  In this paper, we consider the following complex-valued semilinear heat
equation \begin{eqnarray*} \partial_t u = \Delta u + u^p, u \in \mathbb{C},
\end{eqnarray*} in the whole space $\mathbb{R}^n$, where $ p \in \mathbb{N}, p
\geq 2$. We aim at constructing for this equation a complex solution $u = u_1 +
i u_2$, which blows up in finite time $T$ and only at one blowup point $a$,
with the following estimates for the final profile \begin{eqnarray*} u(x,T)
&\sim & \left[ \frac{(p-1)^2 |x-a|^2}{ 8 p |\ln|x-a||}\right]^{-\frac{1}{p-1}},
u_2(x,T) &\sim & \frac{2 p}{(p-1)^2} \left[ \frac{ (p-1)^2|x-a|^2}{ 8p
|\ln|x-a||}\right]^{-\frac{1}{p-1}}\frac{1}{ |\ln|x-a||} , \text{ as } x \to a.
\end{eqnarray*}
  Note that the imaginary part is non-zero and that it blows up also at point
$a$. Our method relies on two main arguments: the reduction of the problem to a
finite dimensional one and a topological argument based on the index theory to
get the conclusion. Up to our knowledge, this is the first time where the
blowup behavior of the imaginary part is derived in multi-dimension.
