I was wondering if anybody could give me some references to already
existing literature for the following open ended problem.

Namely, I am interested in studying the equation of
"complex harmonic oscillator"

$$\ddot{z}(t)+q(t)z(t)=0$$

where $z:\mathbb{R}\to\mathbb{C}$.

The case when $t$ is complex is also interesting and might shed some 
light on to the real case. Assume that the function $q(t)$ is real (for
real time) and strictly decreasing. Actually, without lost of generality
let $q(t)=1/(1+t)$. I have strong numerical evidence for the
following claim. There exist the set of initial conditions such that
$|z|$ is a constant or strictly monotone decreasing (increasing)
functions on the solutions satisfying those initial conditions.

I am considering the following auxiliary function
$\rho(t)=\ln(z\cdot \bar{z})$
the real case but I still can not make a use of the fact that q(t) is
positive and decreasing on $[0,\infty)$.
For the curious, I am essentially play the same kind a game like in the
proof of Sturm-Picone comparison theorem or Poincare-Benedixon theorem.

I started looking into the case of the complex time due to the following
simple observation. Suppose that $z\_1(t)$ and $z\_2(t)$ are two linearly
independent holomorphic solutions of the equation. The Schwarzian
derivative in notation $S$ of the ratio $z\_1/z\_2$ satisfies

$$S(\frac{z\_1}{z\_2})=1/(1+t)$$

Thank you.