The problem with the federal minimum wage debate is we have no good way out of the problem we have created with the minimum wage as it stands now. The current federal minimum wage is set at $7.25 per hour, and has been since 2009. Before then it was set at another fixed amount for a longer period of time.
The pro to linking federal minimum wage to CPI (consumer price index) is the idea of ensuring even the lowest income quintile participates well in the economy, which should be beneficial to a stable aggregate demand and velocity of money. Small increases over time tend to have less impact to labor supply and demand curves than large jumps as well (still there but far less pronounced.) Moderate rises can work on the assumption of dealing with international labor competition.
The negative is no fixed labor costs over a duration of time as applied to products and services in that same economy equates to investment unknowns, and that tends to harm expectations of knowing labor rate (usually the highest cost involved) when it comes to supply and demand for a product or service (and expansion within a market.) Also just about every distortion to labor supply and demand ends up with less paid labor for that product or service in the long term. Advancements, automation, producing more with less labor is a normal expectation now. We have seen this with unions competing with international labor, and we have seen this with certain sectors of our market impacted by higher labor rates because of minimum wage increases (namely large chain retail, fast food, and some aspects of lower cost production.) The last negative is cost of living standards is not uniform across the US, $15 per hour (just as an example) has wildly different spending capabilities when comparing say San Francisco, CA to Birmingham, AL to rural Blur Ridge, GA (again all examples.)
If you linked federal minimum wage to something more like a combination of CPI and productivity you would see it somewhere in the $10 to $13 per hour range.
If you linked federal minimum wage to CPI alone and considering that just about everything in the basket is purchased then you would see it somewhere in the $21 to $23 per hour range.
The debate is always the same, what is compromised to get to those points and stay there based on these factors? (Economists tend to vary a good bit on impact to those very people who work at or just over minimum wage.)