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We study the dynamics of opinion formation in a heterogeneous voter model ona complete graph, in which each agent is endowed with an integer fitnessparameter $k \ge 0$, in addition to its $+$ or $-$ opinion state. The evolutionof the distribution of $k$--values and the opinion dynamics are coupledtogether, so as to allow the system to dynamically develop heterogeneity andmemory in a simple way. When two agents with different opinions interact, their$k$--values are compared and, with probability $p$ the agent with the lowervalue adopts the opinion of the one with the higher value, while withprobability $1-p$ the opposite happens. The winning agent then increments its$k$--value by one. We study the dynamics of the system in the entire $0 \le p\le 1$ range and compare with the case $p=1/2$, which corresponds to thestandard voter model. When $0 \le p < 1/2$, the system approaches exponentiallyfast to the consensus state of the initial majority opinion. The mean consensustime $\tau$ grows logarithmically with the number of agents $N$, and it isgreatly decreased relative to the linear behaviour $\tau \sim N$ found in thevoter model. When $1/2 < p \le 1$, the system initially relaxes to a state withan even coexistence of opinions, but eventually reaches consensus byfinite-size fluctuations. The approach to coexistence is monotonic for $1/2 < p< 0.8$, while for $0.8 \le p \le 1$ there are damped oscillations around thecoexistence value. The final approach to coexistence is approximately a powerlaw $t^{-b(p)}$ in both regimes, where $b$ increases with $p$. Also, $\tau$increases respect to the voter model, although it still scales linearly with$N$. The $p=1$ case is special, with a relaxation to coexistence that scales as$t^{-2.73}$ and a consensus time that scales as $\tau \sim N^\beta$, with$\beta \simeq 1.45$.

Fitness voter model: damped oscillations and anomalous consensus