  We prove that the Gram--Schmidt orthogonalization process can be carried out
in Hilbert modules over Clifford algebras, in spite of the un-invertibility and
the un-commutativity of general Clifford numbers. Then we give two crucial
applications of the orthogonalization method. One is to give a constructive
proof of existence of an orthonormal basis of the inner spherical monogenics of
order $k$ for each $k\in\mathbb{N}.$ The second is to formulate the Clifford
Takenaka--Malmquist systems, or in other words, the Clifford rational
orthogonal systems, as well as define Clifford Blaschke product functions, in
both the unit ball and the half space contexts. The Clifford TM systems then
are further used to establish an adaptive rational approximation theory for
$L^2$ functions on the sphere and in $\mathbb{R}^m.$
