Intuition behind a 0% central/equal-tailed confidence interval At a vague intuitional level, I feel like, if we generated a 0%, two-tailed, truly central (or "equal-tailed") confidence interval (CI), the number it would wrap around would be a median-unbiased estimate of the population parameter in question.  Am I confused, or is this right?  Is there another intuition of it that is better?
I can't seem to find anything about it when I google, but my intuition rests on the idea of the CI as the inversion of a p-value.  I think, this CI would also match one end of a one-sided 50% CI, which would also be the inversion of a 0.5 one-sided p-value.  And a 0.5 one sided p-value is saying "if this were our parameter, then 50% of datasets would be look to be 'higher' and 50% of them 'lower'".  So I feel like my intuition is right, but I can't quite formalise it, or find anything online to confirm or deny this.

A zero-level confidence interval can be seen as an estimator. Indeed, it has been advocated by Skovgaard (1989) "A review of higher-order likelihood inference". Bull. Int. Statist. Inst., 53, 331–351, in a class of two-sided equal-tailed confidence intervals, and is defined as the intersection of all confidence intervals at all levels.
An implementation of this idea needs estimating functions based on pivotal quantities for the parameter of interest. However, unless the pivot is exact and has a symmetric distribution around the true parameter (s.t. in the case of the Student's $t$ statistic), the median unbiasedness property for this kind of estimator does not necessarily hold.
One way to get such an estimator with low median bias is to use a higher-order pivotal quantity such as the $r^*$ of Barndorff- Nielsen (1986) "Inference on full or partial parameters based on the standardized signed log-likelihood ratio". Biometrika, 73, 307-322.
