This video (no need to actually watch it) makes a great point. If we interpret the $f$ in $f(x)$ as a function of time, then the fourier transform of $f$ takes the representation of this function in terms of time to a new representation of $f$ simply in terms of frequency, & this makes sense intuitively since a function can be represented, via Fourier series, as a sum of sine functions, & because any sine function is completely determined by it's amplitude, frequency & phase we should be able to completely characterize the function represented in terms of time by a new function in terms of frequency - if we know the amplitude & phase at every frequency in our sum of sines (& this is found by integrating over all time, across the entire spectrum, in the time representation). From this perspective the utility of complex numbers leaps out since amplitude & phase can be represented in a single complex number...
In other words, take a function, represent it as it's fourier series, decompose the coefficients into of the sines & cosines into their integral representations, do the algebra to reduce this to the Fourier transform representation, then if you wish represent this in terms of complex numbers - this decomposes a function represented in terms of time to a new representation in terms of frequency. This whole process can also be achieved by "pre-multiplying" the function by a complex exponential with a purely imaginary argument and integrating.
From this perspective the Laplace transform is merely just the Fourier transform where the complex exponential also has a real argument.
But I just do not see how that extra step falls out of the development I've written above, I don't see where the real argument rears it's ugly headm or what it means.