The Schrödinger equation describes the behavior of an entity called the wave-function. The wave-function is not real; it does not represent any physical phenomenon. Instead, it represents a mathematical simplification which transforms a heinous differential equation into a merely wretched one that describes far more than the probability distribution we get from $\left<\psi | \psi \right>$.
In some sense, the Schrödinger equation and the wave-function it describes are the result of a square root, but the equation squared is of far less value because it does not allow us to use operators. For example, to find an expectation value of a particular wave-function, we typically find
$$\left< \psi |\mathcal O |\psi \right>$$
where $\psi$ represents the wave-function and $\mathcal O$ represents some operator. For example, if we want to know the expectation value for the momentum of $\left| \psi \right>$, we would choose an appropriate basis for $\left| \psi \right>$ and then find $\left<\psi | p | \psi \right>$.
In differential equations, imaginary coefficients on first order linear equations change exponential behavior into sinusoidal behavior. In the simplest second order linear differential equation, $f(x) + a f''(x) = 0$, complex $a$ changes $f$ from a sinusoid to some product of sinusoidal and exponential functions.