quantum mechanics - Solving the time independent Schrodinger equation: Does a complex solution make sense?

In my notes, I have the Time Independent Schrodinger equation for a free particle

$$\frac{\partial^2 \psi}{\partial x^2}+\frac{p^2}{\hbar^2}\psi=0\tag1$$

The solution to this is given, in my notes, as

$$\Large \psi(x)=C e^\left(\frac{ipx}{\hbar}\right)\tag2$$

Now, since (1) is a second order homogeneous equation with constant coefficients, given the coefficients we have, we get a pair of complex roots:

$$r_{1,2}=\pm \frac{ip}{\hbar}\tag3$$

Thus, the most general solution looks something like:

$$\psi(x)=c_1 \cos \left(\frac{px}{\hbar}\right)+c_2 \sin \left(\frac{px}{\hbar}\right)\tag4$$

However, instead of writing the solution as a cosine plus a sin, the professor seems to have taken a special case of the general solution (with $c_1=1$ and $c_2=i$) and converted the resulting

$$\psi(x)=\cos \left(\frac{px}{\hbar}\right)+ i\sin \left(\frac{px}{\hbar}\right)\tag5$$ into exponential form, using $$e^{i\theta}=\cos \theta + i\sin \theta \tag6$$ to get (2).

The main question I have concerning this is: shouldn't we be going after real solutions, and ignoring the complex ones for this particular situation? According to my understanding $\Psi(x,t)$ is complex but $\psi(x)$ should be real. Thanks in advance.