This question has been crossposted from Math.SE in the hopes that it reaches a larger audience here.

$\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an orientation-reversing diffeomorphism (complex conjugation!), so this is diffeomorphic to the blowup of $\Bbb{CP}^{2n+1}$ at one point.

On the other hand, $\Bbb{CP}^2 \# \Bbb{CP}^2$ does not even support an almost complex structure: Noether's formula demands that its first Chern class $c_1^2 = 2\chi + 3\sigma = 14$, but if $c_1 = ax_1 + bx_2$ (where $x_1, x_2$ generate $H^2$, $x_1^2 = x_2^2$ is the positive generator of $H^4$, and $x_1x_2 = 0$), then $c_1^2 = a^2 + b^2$, and you cannot write $14$ as a sum of two squares.

Using a higher-dimensional facsimile of the same proof, I wrote down a proof here that $\Bbb{CP}^4 \# \Bbb{CP}^4$ does not admit an almost complex structure. The computations using any similar argument would, no doubt, become absurd if I increased the dimension any further.

Can any $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ support an almost complex structure?