What is the significance of the Schrodinger Equation requiring complex solutions but the Klein-Gordon permitting purely real and imaginary solutions? My intuition for the significance of complex solutions in QM initially was that the phase of wavefunction solutions was more 'rotating' than 'oscillating' (e.g. more like a rotating string (spiral) wave than an oscillating up-down wave on a string).
However, the Klein-Gordon equation permits purely real and purely imaginary solutions (as well as complex solutions) which would suggest 'oscillating' rather than 'spiraling' wavefunction solutions can exist, but only for spin 0 particles?
What is the significance of this? Is there any intuition to be gained?
 A: Assuming that we're considering the complex Klein-Gordon (KG) equation (which is equivalent to 2 real KG equations), there are various arguments why the complex KG field is best Fourier expanded in oscillatory exponential phases rather than sine and cosine to reveal the physics, see e.g. this Phys.SE post.
For starters, the interpretation in terms of creation and annihilation operators becomes physically clearer. In particular, the derivation of the Schrödinger equation in a 1-particle sector from the KG equation relies on a factorization of a rest-energy into an oscillatory exponential phase, cf. e.g. this , this & this Phys.SE posts.
As to the complex nature of the wave function in the Schrödinger equation, see e.g. this Phys.SE post.
A: What the Klein-Gordon equation essentially distinguishes from Schroedinger's equation is the fact that it is more (or even 100%) like a field equation:
$$(\Box + m^2)\phi = 0$$
Why is this ? First of all, it is a relativistic equation. In a relativistic theory the concept of a wavefunction no longer works since it is not causal.
In Schroedinger's equation the parameter $t$, the time, plays a special role. In a relativistic theory all coordinates of the spacetime should be treated on the same footing. Therefore $\phi$ should be no longer treated as a function of time which describes possible localization of a single particle, but as a function of all four coordinates: $\phi=\phi(t,x,y,z)$.
In case of the interpretation of $\phi$ as a field a probabilistic interpretation of $\phi$ is no longer possible and no longer necessary.
On the other hand a wavefunction makes only sense being complex in order to describe correctly interferences, however, for the solution of a field equation we ask above all for the number of degrees of freedom. For a very simple system we only need one degree of freedom that $\phi$ might describe, so a real solution could be sufficient. For system with 2 degrees of freedom one would choose a complex solution.
Actually, the Klein-Gordon equation can transformed back into a Schroedinger-like equation in the non-relativistic limit. Each $2^{nd}$ order equation can be transformed into a first order equation in time by the definition of new variables. These are for the Klein-Gordon equation (and here the complex numbers come again in):
$$\theta = \frac{1}{2}\left( \phi + \frac{i}{m} \dot{\phi}\right)$$
and
$$\chi = \frac{1}{2}\left( \phi - \frac{i}{m} \dot{\phi}\right)$$
Using $\theta$ and $\chi$ a 2-component matrix equation can be defined like
$$\phi = \left[ \begin{array}{c} \theta \\ \chi \end{array}\right]$$
fulfilling
$$i\frac{\partial \Phi}{\partial t} = H_0 \Phi$$
$\Phi$ is necessarily complex by virtue of its definition.
However, the Hamiltonian $H_0$ still is not a non-relativistic one, since it is non-hermitean which does not allow for a probability interpretation of $|\Phi|^2$. But the form of the equation suggests that the parameter time $t$ seems to have taken over a special role.
But in order to make it really non-relativistic a "Foldy-Wouthuysen" transformation has to be carried out. Only then we get to the non-relativistic limit and all the familiar properties of a solution of a Schroedinger-like equation emerge. More on this specific topic can be read in Bjorken & Drell, Relativistic Quantum Mechanics Vol. 1.
