Consider a real vector space generated by and . There is an obvious identification of with the complex plane such that and . Define a linear complex structure on by setting and . With the identification mentioned above, since is a complex vector space, can be viewed as a complex vector space, too. Furthermore, the action of can be viewed as multiplication by on but we will see below why this view does not extend further.
Next, we complexify by taking a tensor product with over . We know that (real) dimension of is and it is generated by and . We can view as a complex vector space and, for notational simplicity, write and . Note that over the complex numbers is dimensional and generated by and . However, these are not the “natural” bases to work with as we wil see. Next, we extend (complexify) to get which we will still denote by for notational simplicity. Let and . Now, we see that
This means that is an eigenvector of corresponding to the eigenvalue . Similarly, is an eigenvector corresponding to the eigenvalue . So, the set is an eigenbasis for and it gives us an eigenspace decomposition of . Computing , using this basis, is clearly more convenient and hence, this is a “natural” choice as a basis. Furthermore, from this viewpoint, it is also clear why the action of cannot be viewed as multiplication by any more.