  The theory of Toeplitz quantization presented in our previous paper is
extended and further developed to include diverse and interesting
non-commutative realizations of the classical Euclidean plane. This is done
using Hilbert spaces of entire functions, where polynomials in one complex
variable form a dense subspace. The complex coordinate naturally acts as an
unbounded multiplication operator generating, together with its adjoint, a
highly non-commutative *-algebra of operators. The Toeplitz operators are then
geometrically constructed as special elements from this algebra; they are
associated to the symbols from another quadratic non-commutative algebra, which
is interpretable as polynomials over a plane to be quantized. Such a conceptual
framework promotes interesting non-trivial conditions on the initial scalar
product. These are analyzed in detail. Various illustrative examples are
computed.
