Trans-dimensional Bayesian sampling has been applied to subsurface imaging and other inference problems across the Earth Sciences. A particular style of Markov chain Monte Carlo (McMC) method, known as reversible-jump has been used almost universally in such studies. This algorithm allows sampling across variably dimensioned model parameterizations. However, for practical reasons, it is limited to cases where the number of free parameters differ in a regular sequence between alternate models, usually by addition or subtraction of a single variable. Furthermore, jumps between model dimensions rely on bespoke mathematical transformations, which are bespoke to each class of application. As a result, implementations are dependent on the choice of model parameterization employed. A framework for Trans-conceptual Bayesian sampling, which is a generalization of trans-dimensional sampling, is presented. Trans-C Bayesian sampling allows exploration across a finite, but arbitrary, set of conceptual models, i.e. ones where the number of variables, the type of model basis function, nature of the forward problem, and assumptions on the measurement noise statistics, may all vary independently. The new framework avoids parameter transformations and thereby lends itself to development of automatic McMC algorithms, i.e. where the details of the sampler do not require knowledge of the parameterization. Algorithms implementing Bayesian conceptual model sampling are presented and illustrated with examples drawn from geophysics, using real and synthetic data. Comparison with reversible-jump illustrates that Trans-C sampling produces statistically identical results for situations where the former is applicable, but also allows sampling in situations where Trans-D would be impractical to implement.