Measuring growth rate when values are negative
For negative values alone you can define relative change as:
$$\frac{X_t-X_{t-1}}{|X_{t-1}|}$$
This is quite common way to deal with rates of change when you have negative numbers. However, when the denominator is zero then the growth rate is not defined. This can solve any issue when zero is not a base.
When zero is the base unfortunately there is no way how you can calculate growth rate while keeping the zero there. This is because if you start from zero any change represents $\infty$ increase. Going from 0 to 1 is $\infty$ growth rate as well as going from 0 to 10 or 0 to 1000. You have to either get rid of a zero in some way or another or simply do not calculate growth rate but some other measure of change (e.g. simple change $X_t -X_{t-1}$).

I have already thought of adding an arbitrary but reasonable positive number to data points, to bring everything into a positive domain. But this is a very crude and not an elegant way to do this.

Yes, this is very crude way of doing it but perhaps there are more elegant alternatives. Using the temperature as an example a solution would be to convert degrees Celsius to Kelvin (with $T_{(K)} = T_{(°C)} + 273.15$) which would solve the problem.
Perhaps there is an another way to measure the quantity you want. Otherwise, adding an arbitrary constant would be a solution. It is hard to recommend any specific way without knowing all the details of the research.