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For whatever it's worth: $S^6$ is a ($6$-dimensional) homogenous nearly Kähler manifold. These were classified by Butruille. In particular, they have a canonical almost complex structure. I assume one should be able to prove that this almost complex structure is not a complex structure.

[EDIT, thanks Spiro] Apparently this does not imply that one may conclude that there is no complex structure on $S^6$, so this is not much help.

There are also some interesting uniqueness results for connections on $S^6$ in Fukami and Ishihara's paper that might be helpful for this.

[EDIT, thanks David] For a short time there was a preprint on arXiv claiming that $S^6$ is a complex manifold. See discussion here.

For whatever it's worth: $S^6$ is a ($6$-dimensional) homogenous nearly Kähler manifold. These were classified by Butruille. In particular, they have a canonical almost complex structure. I assume one should be able to prove that this almost complex structure is not a complex structure.

[EDIT, thanks Spiro] Apparently this does not imply that one may conclude that there is no complex structure on $S^6$, so this is not much help.

There are also some interesting uniqueness results for connections on $S^6$ in Fukami and Ishihara's paper that might be helpful for this.

[EDIT, thanks David] For a short time there was a preprint on arXiv claiming that $S^6$ is a complex manifold. See discussion here.

For whatever it's worth: $S^6$ is a ($6$-dimensional) homogenous nearly Kähler manifold. These were classified by Butruille. In particular, they have a canonical almost complex structure. I assume one should be able to prove that this almost complex structure is not a complex structure.

[EDIT, thanks Spiro] Apparently this does not imply that one may conclude that there is no complex structure on $S^6$, so this is not much help.

There are also some interesting uniqueness results for connections on $S^6$ in Fukami and Ishihara's paper that might be helpful for this.

[EDIT, thanks David] For a short time there was a preprint on arXiv claiming that $S^6$ is a complex manifold. See discussion here.

For whatever it's worth: $S^6$ is a ($6$-dimensional) homogenous nearly Kähler manifold. These were classified by Butruille. In particular, they have a canonical almost complex structure. I assume one should be able to prove that this almost complex structure is not a complex structure.

[EDIT, thanks Spiro] Apparently this does not imply that one may conclude that there is no complex structure on $S^6$, so this is not much help.

There are also some interesting uniqueness results for connections on $S^6$ in Fukami and Ishihara's paper that might be helpful for this.

[EDIT, thanks David] For a short time there was a preprint on arXiv claiming that $S^6$ is a complex manifold. See discussion here.

For whatever its worth: $S^6$ is a ($6$-dimensional) homogenous nearly Kähler manifold. These were classified by Butruille. In particular, they have a canonical almost complex structure. I assume one should be able to prove that this almost complex structure is not a complex structure.

[EDIT, thanks Spiro] Apparently this does not imply that one may conclude that there is no complex structure on $S^6$, so this is not much help.

There are also some interesting uniqueness results for connections on $S^6$ in Fukami and Ishihara's paper that might be helpful for this.

[EDIT, thanks David] For a short time there was a preprint on arXiv claiming that $S^6$ is a complex manifold. See discussion here.

For whatever it's worth: $S^6$ is a ($6$-dimensional) homogenous nearly Kähler manifold. These were classified by Butruille. In particular, they have a canonical almost complex structure. I assume one should be able to prove that this almost complex structure is not a complex structure.

[EDIT, thanks Spiro] Apparently this does not imply that one may conclude that there is no complex structure on $S^6$, so this is not much help.

There are also some interesting uniqueness results for connections on $S^6$ in Fukami and Ishihara's paper that might be helpful for this.

[EDIT, thanks David] For a short time there was a preprint on arXiv claiming that $S^6$ is a complex manifold. See discussion here.