What are imaginary signals?
How can a signal be imaginary?

It's not imaginary in the everyday sense of meaning something that doesn't exist.
&quot;Real&quot; and &quot;imaginary&quot; have technical meanings in signal processing, and more widely in mathematics, that are different from their everyday meanings.
In signal processing &quot;real&quot; denotes a signal component that is in phase with (i.e. has a 0° phase shift from) some reference signal.  Imaginary denotes a signal component that is in quadrature with (i.e. has a 90° phase shift from) the same reference signal. The reference signal can come from a local oscillator. (In DSP or SDR equipment the local oscillator might be a mathematical representation of one, rather than an analog circuit.) In a receiver we might seek to synchronise the reference signal to the received signal.
In signal processing equipment we also refer to real and imaginary as I and Q for in-phase and quadrature, respectively. Some equipment has sections which are divided into separate I and Q channels. You can measure the signals flowing through a Q (imaginary) channel, implemented in hardware, and find they are just as real (in the everyday sense) as the signals flowing through the I (real) channel.
The term &quot;imaginary&quot; was originally coined in the 17th century by René Descartes as a derogatory term and has been confusing students ever since.
For why $e^{j \omega t} = \cos  \omega t + j \sin \omega t $ see Euler's formula. Mathematicians use $i$ to represent the imaginary unit. Electrical engineers use $j$ to represent the imaginary unit because we use $i$ to represent (instantaneous) current.  You can use either representation but please choose one and stick with it.
(This question was originally posted in the amateur radio SE group. This answer was drafted as a reply there and leans towards radio usage.)