It looks that way.
The question that remains after competing bills passed each house in the state legislature on May 4 is exactly where in March the primary ends up in 2020. The Assembly bill -- AB 84 -- is the more straightforward of the two. It shifts a consolidated California primary back into the first Tuesday (after the first Monday) in March position the Golden state primary occupied during the 2000 and 2004 cycles.
Alternatively, the Senate version -- SB 568 -- is a bit more complex. On the surface, it stakes out a less ambitious, third Tuesday in March spot on the calendar. However, it also gives the California governor the option of bumping the primary up to an even earlier position. Now, given current national party timing penalties on presidential primaries, anything earlier than the first Tuesday in March is a non-starter. That gubernatorial power is rendered mostly powerless; except granting the executive branch the ability to move the primary up as early as the date called for in the Assembly legislation.
While it is true that a version of this March primary legislation has passed each chamber of the California state legislature, one version obviously still has to pass the opposite house and be signed into law. The first of those seems more likely than not, but the gubernatorial signature is not assured (though admittedly likely given the synchronicity of an open 2020 Democratic presidential nomination and Democrats' unified control in the Golden state).
The likelihood of a move, then is quite high. That carries with it a couple of additional notes worth mentioning here.
First, this move would be on the early side. While some states schedule their presidential primaries for the next cycle before the midterm year, it is exception rather than rule. The vast majority states hold off on acting until the timing of the presidential primary is actually on most legislators radar. Occasionally, that is immediately after a presidential election year, but more often than not, the urgency is higher after the midterms and more importantly after the national parties have set their delegate selection rules for the upcoming cycle.
Since this California move would be made earlier than is typical, the legislature would theoretically have the ability to move again. That is probably not likely this time around, but it should be noted that California shifted from March to June after 2004 before moving again -- from June to February -- ahead of the 2008 cycle. All the incentives are to move earlier, and in this 2020 case, California will already be as early as it possibly can be. Unless the national parties seek to penalize March primaries, then, California would be unlikely to pull up the tent stakes and move again; this time to a later date.
Additionally, a California push into March has a certain gravitational pull to it. In delegates terms, the cache of delegates in California represented nearly seven percent of the total number of Republican delegates in 2016 and nearly 11.5 percent on the Democratic side. That is a level of delegate-richness that brings other states along for the ride. That happened in 1996, 2000 and 2008 -- all cycles that witnessed California shifting to an earlier primary date.
While there are only a handful of states with unified Democratic control now -- and thus have less motivation to move -- that may be different after the 2018 midterms. That is still a way off, but bears mention in this context. California may be alone in moving now, but would potentially pull many of those northeastern, mid-Atlantic states that shifted from February to much later dates in 2012 back into March, further frontloading the calendar.
Most of this is a story for 2019. But for now, the momentum is behind a California push into some Tuesday position in March.