  In this paper, we consider the following semi-linear complex heat equation
\begin{eqnarray*} \partial_t u = \Delta u + u^p, u \in \mathbb{C}
\end{eqnarray*} in $\mathbb{R}^n,$ with an arbitrary power $p,$ $ p > 1$. In
particular, $p$ can be non integer and even irrational. We construct 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 $a.$ Moreover, we also describe the
asymptotics of the solution by the following final profiles: \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||} > 0 , \text{ as } x \to a. \end{eqnarray*}
  In addition to that, since we also have $u_1 (0,t) \to + \infty$ and
$u_2(0,t) \to - \infty$ as $t \to T,$ the blowup in the imaginary part shows a
new phenomenon unkown for the standard heat equation in the real case: a non
constant sign near the singularity, with the existence of a vanishing surface
for the imaginary part, shrinking to the origin. In our work, we have succeeded
to extend for any power $p$ where the non linear term $u^p$ is not continuous
if $ p$ is not integer. In particular, the solution which we have constructed
has a positive real part. We study our equation as a system of the real part
and the imaginary part $u_1$ and $u_2$. Our work 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.
