  This paper considers the extension of classical Lagrange interpolation in one
real or complex variable to "polynomials of one quaternionic variable". To do
this we develop some aspects of the theory of such polynomials. We then give a
number of related multivariate polynomial interpolation schemes for
${\mathbb{R}}^4$ and ${\mathbb{C}}^2$ with good geometric properties, and some
aspects of least interpolation and of Kergin interpolation.
