Complex nature of chirped signal
Depending on the context, the use of the complex form could be for mathematical convenience or for a no-kidding need for both real and imaginary parts.
When you factor the expression, you get
$$u(t) = e^{{\eta}t^2}e^{j{\beta}t^2}$$
Where the first exponential is a generic magnitude envelope, in this case Gaussian. The second exponential is the chirp itself and is where all the action is.
So to simplify things, lets assume that the envelope is ideal so that the signal is just the chirp
$$u(t) = e^{j{\beta}t^2}$$
To view the chirp, you can take either the real or imaginary part, and it looks something like this

In a system that does not use I/Q, the real part is what you would expect to transmit as your waveform. The type of system will determine whether using a real signal or some type of I/Q is best. I'm going to use a radar example here.
In frequency-modulated continuous wave (FMCW) radars, a real chirp like the one above is transmitted and has the form
$$x(t) = cos({{\beta}t^2})$$
Which is just the real part of the complex form. It is received after a delay and mixed with itself, and without going into the mixing process, produces a single frequency sinusoid that can be used to determine range. Here, using a real part only is practical. Using I/Q in FMCW is also beneficial (SNR improvement), but is not usually necessary and many systems do not use it.
Another type of radar, pulse-Doppler, benefits greatly from using the complex form. The same chirp is considered, except now the imaginary version is used. This is important because pulsed-Doppler radars usually operate on performing pulse compression, which is just correlating the transmitted waveform with the received one.
The autocorrelation of a complex chirp looks like

Using a complex waveform allows us to mix our signals to baseband, which give the classic autocorrelation responses we expect without additional mixing and filtering.