  Let $K$ be an imaginary quadratic field. In this article, we study the
eigenvariety for $\mathrm{GL}_2/K$, proving an \'etaleness result for the
weight map at non-critical classical points and a smoothness result at
base-change classical points. We give three main applications of this; let $f$
be a $p$-stabilised newform of weight $k \geq 2$ without CM by $K$. Suppose $f$
has finite slope at $p$ and its base-change $f_{/K}$ to $K$ is $p$-regular.
Then: (1) We construct a two-variable $p$-adic $L$-function attached to
$f_{/K}$ under assumptions on $f$ that conjecturally always hold, in particular
with no non-critical assumption on $f/K$. (2) We construct three-variable
$p$-adic $L$-functions over the eigenvariety interpolating the $p$-adic
$L$-functions of classical base-change Bianchi cusp forms. (3) We prove that
these base-change $p$-adic $L$-functions satisfy a $p$-adic Artin formalism
result, that is, they factorise in the same way as the classical $L$-function
under Artin formalism.
  In an appendix, Carl Wang-Erickson describes a base-change deformation
functor and gives a characterisation of its Zariski tangent space.
