In Arakelov geometry, the conventional wisdom is that the ``closed fibre at $\infty$'' should be viewed as totally degenerate. This is the extreme opposite of smoothness. A visualization in the case of a curve is proposed in:

Yu. Manin: Three-dimensional hyperbolic geometry as $\infty$-adic Arakelov geometry, *Inventiones*, 1991.

This was inspired by Mumford's work on $p$-adic Schottky groups, extending to higher genus the rigid analytic picture of the Tate curve.

This work of Consani and Marcolli could be also helpful:

C. Consani, M. Marcolli: Noncommutative geometry, dynamics, and $\infty$-adic Arakelov geometry, *Selecta Math.* 2004.

The Tate curve may be enough to convince you of such a point of view. Analytically, over $\mathbb{C}_p$ or $\mathbb{C}$, a $\mathbb{G}_m$-reduction elliptic curve or a complex elliptic curve are, respectively, both a quotient of $\mathbb{G}_m$ by a one-dimensional group of periods. There is little similarity with an elliptic scheme over $\mathbb{Z}_p$. In the Tate curve, making increasingly ramified extension of the base field, the closed fibre of the minimal desingularization is a Neron polygon of projective lines, whose group of irreducible components converges to a circle $\mathbb{R}/\mathbb{Z}$: a tropical elliptic curve, and the skeleton of the Berkovich analytification. In contrast, for an elliptic curve over $\mathbb{Z}_p$  the analytification is contractible, giving the wrong Betti number in comparison to what happens Archimedeanly. This consideration extends to curves of higher genus $g$: the first Betti number of the analytification is $\leq g$, with equality if and only if the curve is totally degenerating.

The dynamics of rational maps illuminate another, though related, point. If the map has a good reduction at $p$ then its dynamics is completely predictable: the Julia set has no $\mathbb{C}_p$-points. But the Julia sets of complex dynamics are always non-empty, and they are typically quite intricate. 

Finally, we may compare the geometry or dynamics over $\mathbb{Q}$ with the corresponding situation over the rational function field $k(t)$. If in the function field model we have a rational map over $k(t)$ with everywhere potential good reduction, then the map is isotrivial. The irrelevance of isotriviality for algebraic varieties or rational maps over number fields (as in Bogomolov's problem, Lehmer's problem, the $abc$ conjecture) leads us to admit the degeneration of at least some of the fibres: the Archimedean ones. If, for instance, we took the map $z \mapsto z^2$ over $\mathbb{Q}$ to be modeled by the squaring map over $k(t)$, which is isotrivial, the Lehmer problem (on the spectral gap of the dynamical Mahler measure) has a trivial, affirmative answer. Not so if we admit a  non-isotrivial map: conceivably, this will have the same answer and comparable difficulty as the classical Lehmer problem. In the case of $z \mapsto z^2$, a more faithful model for the Lehmer question is not the squaring map of $\mathbb{F}_q(t)$, but the map $c(z) = tz + z^q$, which is the isogeny of multiplication by $t$ on the Carlitz module over $\mathbb{F}_q(t)$. In either case we have exactly the same progress on Lehmer's question, namely, Dobrowolski's bound; and the map $c$ has a bad reduction at the ``infinite place.''

*Added.* Since the question concerned an arbitrary hermitian metric, I felt I should add a pointer to Chambert-Loir and Ducros's work on Chern forms and their associated measures: *Formes differentielles reelles et courants sur les espaces de Berkovich*. Arakelov geometry considers line bundles $\bar{L} = (L,\|
\cdot\|)$ with a *smooth* hermitian metric. The Chern (curvature) form $c_1(
\bar{L})$ is then a smooth $(1,1)$-form, and its determinant (or self intersection) is a full support measure on $X(\mathbb{C}) = X_{/\mathbb{C}}^{\mathrm{an}}$. There are a corresponding form and measure on $X_{/\mathbb{C}_p}^{\mathrm{an}}$, where the $p$-adic metric in $L$ is taken to arise from the integral model of $L$ over $\mathbb{Z}_p$. Now if $X$ has a good reduction at $p$, the obtained measure is the Dirac mass charging a single point of the Berkovich space. This is nothing like the Archimedean case with a $C^{\infty}$ metric, but if $X$ has a sufficiently bad reduction at $p$ then the measure may likewise have a full dimensional support.