Are there analogues to conformal mapping in 3 dimensions?

I have a specific example I am trying to solve.. Laplace's equation in 3D with slightly complicated rectilinear boundaries. (Think of solving a harmonic function over a 3D boundary which is a cube but with a sub-cube "bitten" out of one corner.)

Laplace's equation is still valid under conformal transformations, so for example in 2D I could take a square domain with a subsquare bitten out of a corner, and apply an inverse tranformation like some of these and solve the equation in a simple square domain.

Are there similar conformal-like transformations in 3D? Perhaps they wouldn't be called conformal maps, but maybe something exists which would work similarly for my 3D Laplace equations.