For algebraic numbers this means that the exponent $2+\varepsilon$ in Roth's theorem can be reduced to $2$. For quadratic irrationalities this holds with the uniform constant $c = 1/\sqrt{5}$; google "Lagrange spectrum." As far as I know it is widely believed that this may not happen for any algebraic number of degree $> 2$, although Serge Lang has suggested that a milder improvement in Roth's theorem, whereby $q^{2+\varepsilon}$ gets replaced with $(q\log{q})^2$, is always possible. (whichThis is wide openopen; there is an analogous statement known to hold true in Nevanlinna theory). Certainly no algebraic number of degree $> 2$ is known to have unbounded partial fractions coefficients, but worse, it does not appear to be even known that there exists an algebraic number having this property.
But now there is something even more surprisingcurious about the implication of such examples for Roth's theorem over $\mathbb{Q}(i)$ (diophantine approximations by Gaussian numbers). I realize this just now after looking up Hensley's very interesting paper [2] (from 2006), and I wonder why this point isn't brought up in the literature on complex continued fractions.
While the convergents $p_n/q_n \in \mathbb{Q}(i)$ in Hurwitz's expansion no longer exhaust all good $\mathbb{Q}(i)$-rational approximations to $\alpha \in \mathbb{C}$ (as they do in the case over $\mathbb{Q}$), Theorem 1 in [2] shows that up to a multiplicative constant, they still give the best $\mathbb{Q}(i)$-rational approximations provided that the successive ratios $q_{n+1}/q_n$ are bounded. This boundedness assumption is satisfied for for Hensley's number $\sqrt{2} + i\sqrt{5}$ (see observation (3) on page 12 in loc.cit.), and so is I believe for all the examples previously quoted from [3]. ConsequentlyAs a result, for those numbers (algebraic of arbitrarily high relative degree over $\mathbb{Q}(i)$), the exponent $2+\varepsilon$ in Roth's theorem rel. $\mathbb{Q}(i)$ (stated above) can be reduced to $2$, and this result is moreover effective:
Allow me to record severala few