Here is an illustration of how bias might arise in conclusions, and why it may not be the full story. Suppose you have a sequential trial of a drug which is expected to have a positive (+1) effect but may have a negative effect (-1). Five guinea pigs are tested one after the other. The unknown probability of a positive outcome in a single case is in fact $\frac{3}{4}$ and a negative outcome $\frac{1}{4}$.
So after five trials the probabilities of the different outcomes are
Outcome Probability
+5-0 = +5 243/1024
+4-1 = +3 405/1024
+3-2 = +1 270/1024
+2-3 = -1 90/1024
+1-4 = -3 15/1024
+0-5 = -5 1/1024
so the probability of a positive outcome overall is 918/1024 = 0.896, and the mean outcome is +2.5. Dividing by the 5 trials, this is an average of a +0.5 outcome per trial.
It is the unbiased figure, as it is also $+1\times\frac{3}{4}-1\times\frac{1}{4}$.
Suppose that in order to protect guinea pigs, the study will be terminated if at any stage the cumulative outcome is negative. Then the probabilities become
Outcome Probability
+5-0 = +5 243/1024
+4-1 = +3 324/1024
+3-2 = +1 135/1024
+2-3 = -1 18/1024
+1-2 = -1 48/1024
+0-1 = -1 256/1024
so the probability of a positive outcome overall is 702/1024 = 0.6855, and the mean outcome is +1.953. If we looked the mean value of outcome per trial in the previous calculation, i.e. using $\frac{+5}{5}$, $\frac{+3}{5}$, $\frac{+1}{5}$, $\frac{-1}{5}$, $\frac{-1}{3}$ and $\frac{-1}{1}$ then we would get +0.184.
These are the senses in which there is bias by stopping early in the second scheme, and the bias is in the predicted direction. But it is not the full story.
Why do whuber and probabilityislogic think stopping early should produce unbiased results? We know the expected outcome of the trials in the second scheme is +1.953. The expected number of trials turns out to be 3.906. So dividing one by the other we get +0.5, exactly as before and what was described as unbiased.