This question has now been crossposted to MathOverflow, in the hopes that it reaches a larger audience there.
CP2n+1#CP2n+1 supports a complex structure: CP2n+1 has an orientation-reversing diffeomorphism (complex conjugation!), so this is diffeomorphic to the blowup of CP2n+1 at one point.
On the other hand, CP2#CP2 does not even support an almost complex structure: Noether’s formula demands that its first Chern class c21=2χ+3σ=14, but if c1=ax1+bx2 (where x1,x2 generate H2, x21=x22 is the positive generator of H4, and x1x2=0), then c21=a2+b2, 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 CP4#CP4 does not admit an almost complex structure. The computations using any similar argument would, no doubt, become absurd if I increased the dimension any more.
Can any CP2n#CP2n support an almost complex structure?
The question has been answered in the crosspost to mathoverflow by Panagiotis Konstantis. I copy this answer here to close the question.
The m-fold connected sum m#CP2n admits an almost
complex structure if and only if m is odd, as we show in our recent
[preprint]. (By the way, thanks to Mike for this interesting
question, which motivated us to write the paper!)
Here’s a brief summary of the proof’s idea. Our main tool is a result
by Sutherland resp. Thomas from the 60s which tells us when a stable
almost complex structure is induced by an honest almost complex
structure: this is the case iff its top Chern class equals the Euler
class of the manifold.
As the connected sum of manifolds admitting a stable almost complex
structure admits one as well, we certainly have stable almost complex
structures on m#CP2n at our disposal, and we can
understand the full set of all such structures by explicitly
determining the kernel of the reduction map from complex to real
K-theory. We then compute the top Chern class of all these structures:
luckily for us, it turns out that in order to show the non-existence
of almost complex structures for even m, it suffices to compute its
value modulo 4 and compare it to the Euler characteristic of m#>CP2n. For odd m, we explicitly find a stable almost
complex structure for which the criterion above is satisfied.