What is the minimum number of time points recommended for accurate instantaneous phase estimates using the Hilbert transform
Forgive me for not having a numerical answer to the question. But here are a few things to think about. The answer depends on how the Hilbert transform (HT) is implemented. If the HT is implemented using an $N$-point tapped-delay line filter-like block diagram the first and last $N$-1 output samples are incorrect. So in this situation four input times samples is definitely not enough samples to produce useful results.

If HT is implemented by way of a forward FFT, zero the neg-frequency spectral components, and inverse FFT (as is done in Matlab) you have a better chance of computing meaning results. However, in this second implementation the input time sequence must be long enough to contain at least one cycle of all the spectral components of your input signal in order to compute somewhat meaningful results. It seems to me that four times samples is too short to reliably estimate any signal parameters of most real-world (information-carrying) signals.